Extensions and Applications of Locally Solid Convergence Structures

Abstract

Locally solid convergence structures provide a unifying framework for both topological and non-topological convergences in vector lattice theory. In this paper, we explore various extensions and applications of locally solid convergence structures. We characterize unbounded locally solid convergences in different spaces, establish connections with bornological convergences, and investigate their applications in functional analysis. Additionally, we generalize these structures to non-Archimedean vector lattices and compare them with traditional topological frameworks. Finally, we develop applications in fixed point theory and operator spaces. Our results contribute to a deeper understanding of the interplay between different types of convergence structures in mathematical analysis.

1. Introduction

The theory of locally solid topologies in vector lattice theory has long served as a cornerstone in functional analysis, offering a natural setting to study the interplay between order and convergence [1,2,3]. Traditionally, these topologies have been essential for exploring properties such as order continuity and duality in ordered vector spaces. However, over the past few decades, it has become increasingly apparent that many classical convergence phenomena in vector lattices are inherently non-topological. This realization has motivated the development of the more general framework of locally solid convergence structures, which unifies both topological and non-topological modes of convergence [4,5,6].

Early foundational work rigorously defined locally solid topologies and established their fundamental properties in vector lattices [1]. These studies demonstrated that local solidity—the requirement that the lattice operations remain continuous with respect to the convergence structure—is critical for capturing the intrinsic order-based behavior of these spaces. Building upon these insights, subsequent research extended the classical framework to include non-topological convergences. In particular, the introduction of unbounded modifications has provided a powerful tool for analyzing convergence phenomena that lie beyond the reach of traditional topological methods [7,8].

Parallel to these developments, the concept of bornological convergence has emerged as a significant refinement. By incorporating bornologies—which formalize the notion of boundedness in vector lattices—into the locally solid framework, researchers have developed bounded modifications that capture convergence behavior on bounded subsets [9,10,11]. These advancements have not only deepened our understanding of the structure of vector lattices but have also led to important applications in duality theory and the analysis of operator spaces [12,13,14].

More recently, the literature has witnessed further generalizations of locally solid convergence structures to non-Archimedean settings. In these contexts, where the classical Archimedean property is relaxed, the convergence framework has been adapted to accommodate ultrametric and p-adic norms [13,15]. Such extensions are particularly relevant for p-adic analysis and ultrametric functional analysis, opening new avenues for exploring convergence phenomena in settings where traditional norms and topologies are inadequate. Additionally, Choquet modifications have been proposed to incorporate capacity-based convergence criteria, thereby capturing even subtler aspects of convergence in vector lattices.

In this paper, we provide a comprehensive study of these extended locally solid convergence structures. Our objective is to characterize various modifications—such as unbounded, bornological, and Choquet modifications—and establish rigorous connections with classical topological frameworks. We also explore applications of these theories in functional analysis, operator theory, and fixed point theory. Through an in-depth review of the literature and a detailed examination of both classical and modern approaches, we aim to contribute to a deeper understanding of convergence phenomena in vector lattices and stimulate further research in this evolving area.

2. Preliminaries

In this section, we provide fundamental definitions and results related to locally solid convergence structures, which will be used in subsequent sections. We follow the standard terminology and notation from vector lattice theory and convergence spaces [1,5,9,16].

A convergence structure on a set X is a relation between nets $\left(x_{\alpha }\right)$ in X and points of X satisfying certain axioms. The following definitions extend the concept of topological convergence to more general settings.

Definition 1 

(Net Convergence [5]). A net $\left(x_{\alpha }\right)$ in a set X is said to converge to $x\in X$ under a convergence structure λ, denoted $x_{\alpha }\stackrel{\lambda }{\to }x$, if for every set U in some neighborhood system of x, there exists ${\alpha }_0$ such that $x_{\alpha }\in U$ for all $\alpha \ge {\alpha }_0$.

Definition 2 

(Filter Convergence [9]). A filter $\mathcal{F}$ on X is said to converge to $x\in X$ if every neighborhood of x contains a set from $\mathcal{F}$.

Definition 3 

(Locally Solid Convergence [6]). A convergence structure λ on a vector lattice X is called locally solid if it satisfies the following conditions:

1. 

If $x_{\alpha }\stackrel{\lambda }{\to }0$ and $|y_{\alpha }| \le |x_{\alpha }|$ for all α, then $y_{\alpha }\stackrel{\lambda }{\to }0$.

2. 

If $x_{\alpha }\stackrel{\lambda }{\to }0$ and $y_{\alpha }\stackrel{\lambda }{\to }0$, then $x_{\alpha } + y_{\alpha }\stackrel{\lambda }{\to }0$.

Lemma 1 

(Uniqueness of Limits in $w\lambda$-Convergence). Let $(X,\lambda )$ be a locally solid convergence vector lattice and suppose that a net $\left(x_{\alpha }\right)$ in X converges to both x and y in the $w\lambda$-convergence structure, that is,

$$x_{\alpha }\stackrel{w\lambda }{\to }x\mathit{and}{x}_{\alpha }\stackrel{w\lambda }{\to }y.$$

Then, $x=y$.

Proof. 

By definition of $w\lambda$-convergence, for every $u\in X_{+}$, we have

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathrm{and}|x_{\alpha }-y| \wedge u\stackrel{\lambda }{\to }0.$$

Using the triangle inequality and the solidity of $\lambda$, we obtain

$$|x-y| \wedge u \le |x_{\alpha }-x| \wedge u + |x_{\alpha }-y| \wedge u.$$

Taking the limit along the net, it follows that

$$|x-y| \wedge u\stackrel{\lambda }{\to }0\mathrm{for}\mathrm{every}u\in X_{+}.$$

Since the set $\{u\in X_{+}:u>0\}$ is directed and the only element that satisfies

$$|z| \wedge u=0\mathrm{for}\mathrm{all}u\in X_{+}$$

is $z=0$, we conclude that $|x-y|=0$, i.e., $x=y$.  □

Lemma 2 

(Absorption Property of Solid Neighborhoods). Let $(X,\lambda )$ be a locally solid convergence vector lattice and let U be a solid λ-neighborhood of 0. Then, for every bounded subset $B\subset X$ and every $x\in X$, there exists a scalar $\alpha >0$ such that for all $y\in B$, if

$$|y| \le \alpha |x|,$$

then $y\in U$.

Proof. 

Since U is a solid neighborhood of 0, by definition for any $z\in X$ satisfying $|z| \le |w|$ with $w\in U$, it follows that $z\in U$. In particular, for each bounded set B, there exists a scalar $\alpha >0$ (depending on B and U) such that $B\subset \alpha U$, that is, for every $y\in B$, there exists $w\in U$ with $|y| \le \alpha |w|$. By the solidity of U, this implies that $y\in U$.  □

Lemma 3 

(Existence of a Regulating Element $u_0$). Let $(X,\lambda )$ be a locally solid convergence vector lattice and let U be a solid λ-neighborhood of 0. Assume that $I\subset X$ is an ideal such that $I^{+}$ is nonempty. Then, there exists an element $u_0\in I^{+}$ such that for every $y\in X$, if

$$|y| \wedge u_0\in U,$$

then $y\in U$.

Proof. 

Since U is a solid neighborhood of 0, by the absorption property, there exists a scalar $\alpha >0$ such that for any $y\in X$,

$$|y|\in U\mathrm{whenever}|y| \le \alpha w$$

for some $w\in U$. In particular, because $I^{+}$ is nonempty, we can select any $u\in I^{+}$. Then, by scaling u, we can choose

$$u_0=min\{\alpha u,u\},$$

which belongs to $I^{+}$ (since I is an ideal and is closed under scalar multiplication and taking minimums in the order structure). With this choice, for any $y\in X$, if

$$|y| \wedge u_0\in U,$$

then the solidity of U ensures that $y\in U$.  □

Definition 4 

(Unbounded Convergence [7,8]). Let X be a vector lattice and let λ be a locally solid convergence structure. The unbounded modification of λ, denoted $u\lambda$, is defined by

$$x_{\alpha }\stackrel{u\lambda }{\to }x \mathit{if} \mathit{and} \mathit{only} \mathit{if} |x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0 \mathit{for} \mathit{all}u\in X_{+}.$$

Remark 1 

(On the Use of $I^{+}$). In our definition of unbounded convergence, for an ideal $I\subset X$, we define

$$I^{+}:=I\cap X_{+},$$

where $X_{+}$ denotes the positive cone of the vector lattice X. Then, the unbounded modification of λ, denoted by $u_I\lambda$, is given by

$$x_{\alpha }\stackrel{u_I\lambda }{\to }x\mathit{if}\mathit{and}\mathit{only}\mathit{if}|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathit{for}\mathit{every}u\in I^{+}.$$

This formulation is in line with standard definitions in the literature (see, e.g., [17] and related references in [18,19]). The use of $I^{+}$ serves to restrict the convergence criteria to elements that are both in the ideal I and positive, which is essential for capturing the order structure in locally solid convergence spaces. We include this remark to provide additional context and clarify that our approach is consistent with established methods in the study of vector lattices and their convergence properties.

Lemma 4 

(Properties of Unbounded Convergence [20]). If λ is a locally solid convergence structure on a vector lattice X, then its unbounded modification $u\lambda$ satisfies the following:

1. 

$u\lambda$ is always locally solid.

2. 

If λ is topological, then $u\lambda$ is also topological.

3. 

If X has a strong unit, then λ and $u\lambda$ coincide.

Definition 5 

(Bornology [9]). A bornology on a vector lattice X is a collection $\mathcal{B}$ of subsets of X, called bounded sets, satisfying the following:

1. 

$\mathcal{B}$ covers X.

2. 

$\mathcal{B}$ is closed under finite unions.

3. 

If $A\in \mathcal{B}$ and $B\subseteq A$, then $B\in \mathcal{B}$.

Theorem 1 

(Bounded Modification [10]). Let λ be a locally solid convergence structure. The bounded modification of λ, denoted $b\lambda$, is defined by

$$x_{\alpha }\stackrel{b\lambda }{\to }x \mathit{if} \mathit{and} \mathit{only} \mathit{if} \mathit{there} \mathit{exists} \mathit{a} \mathit{bounded} \mathit{set} \mathit{B} \mathit{such} \mathit{that} x_{\alpha }-x\in B \mathit{and} x_{\alpha }\stackrel{\lambda }{\to }x.$$

Then, $b\lambda$ is the finest locally solid convergence agreeing with λ on bounded sets.

Theorem 2 

(Minimality [21]). Let X be a vector lattice. There exists a minimal Hausdorff locally solid convergence structure ${\lambda }_{min}$, satisfying the following:

1. 

If λ is any Hausdorff locally solid convergence, then ${\lambda }_{min} \le \lambda$.

2. 

The adherence of any ideal is preserved under ${\lambda }_{min}$.

These definitions and results provide the foundation for our study of extended locally solid convergence structures in the subsequent sections.

3. Characterization of Unbounded Locally Solid Convergences

In this section, we extend the theory of locally solid convergence structures by providing a deeper analysis of unbounded modifications and their implications. We introduce new definitions and results that enhance the understanding of these structures in vector lattices.

Definition 6 

(Generalized Unbounded Convergence). Let X be a vector lattice with a locally solid convergence structure λ. The generalized unbounded modification, denoted $u_{\mathcal{I}}\lambda$, is defined for an ideal $\mathcal{I}\subseteq X$ as follows:

$$x_{\alpha }\stackrel{u_{\mathcal{I}}\lambda }{\to }x \mathit{if} \mathit{and} \mathit{only} \mathit{if} |x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0 \mathit{for} \mathit{all} u\in {\mathcal{I}}_{+}.$$

This definition extends classical unbounded convergence by allowing different levels of control over the choice of the regulating ideal.

Lemma 5 

(Triangle Inequality Bound in Locally Solid Convergence). Let $\left(x_{\alpha }\right)$ and $\left(y_{\alpha }\right)$ be nets in a vector lattice X equipped with a locally solid convergence structure λ. Suppose that for every $u\in X_{+}$,

$$|x_{\alpha }| \wedge u\stackrel{\lambda }{\to }0\mathit{and}|y_{\alpha }| \wedge u\stackrel{\lambda }{\to }0.$$

Then,

$$|x_{\alpha } + y_{\alpha }| \wedge u\stackrel{\lambda }{\to }0.$$

Proof. 

By the triangle inequality in vector lattices and the solidity of the convergence structure, we have for every $u\in X_{+}$

$$|x_{\alpha } + y_{\alpha }| \wedge u \le (|x_{\alpha }| \wedge u) + (|y_{\alpha }| \wedge u).$$

Since both $(|x_{\alpha }| \wedge u)$ and $(|y_{\alpha }| \wedge u)$ converge to 0 in $\lambda$, their sum also converges to 0. Thus,

$$|x_{\alpha } + y_{\alpha }| \wedge u\stackrel{\lambda }{\to }0.$$

 □

Proposition 1 

(Properties of $u_{\mathcal{I}}\lambda$). Let X be a vector lattice with a locally solid convergence λ and let $\mathcal{I}$ be an ideal in X. Then,

1. 

$u_{\mathcal{I}}\lambda$ is always locally solid.

2. 

If λ is Hausdorff, then $u_{\mathcal{I}}\lambda$ is Hausdorff if and only if $\mathcal{I}$ is order-dense.

3. 

If X is order-complete, then $u_{\mathcal{I}}\lambda$ preserves completeness.

Proof. 

We prove the three claims with the following:

(1) Let $\left(x_{\alpha }\right)$ be a net in X with

$$x_{\alpha }\stackrel{u_I\lambda }{\to }0$$

and let $\left(y_{\alpha }\right)$ be another net satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha .$$

Then, for every $u\in I_{+}$, we have

$$|y_{\alpha }| \wedge u \le |x_{\alpha }| \wedge u.$$

Since $|x_{\alpha }| \wedge u\stackrel{\lambda }{\to }0$ by hypothesis and because $\lambda$ is locally solid (i.e., domination preserves convergence), it follows that

$$|y_{\alpha }| \wedge u\stackrel{\lambda }{\to }0.$$

Thus, $y_{\alpha }\stackrel{u_I\lambda }{\to }0$, showing that $u_I\lambda$ satisfies the solidity condition.

Now, assume that $\left(x_{\alpha }\right)$ and $\left(y_{\alpha }\right)$ are two nets in X with

$$x_{\alpha }\stackrel{u_I\lambda }{\to }0\mathrm{and}{y}_{\alpha }\stackrel{u_I\lambda }{\to }0.$$

Then, for every $u\in I_{+}$,

$$|x_{\alpha } + y_{\alpha }| \wedge u \le \left(|x_{\alpha }| + |y_{\alpha }|\right) \wedge u \le |x_{\alpha }| \wedge u + |y_{\alpha }| \wedge u.$$

Since both $|x_{\alpha }| \wedge u$ and $|y_{\alpha }| \wedge u$ converge to 0 in $\lambda$, it follows by the continuity of addition in the locally solid structure that

$$|x_{\alpha } + y_{\alpha }| \wedge u\stackrel{\lambda }{\to }0.$$

Hence, $x_{\alpha } + y_{\alpha }\stackrel{u_I\lambda }{\to }0$. This verifies that $u_I\lambda$ is indeed locally solid.

(2) (a) Necessity. Assume that $u_I\lambda$ is Hausdorff. Suppose, for contradiction, that I is not order-dense in X. Then, there exists an element $x\in X_{+}$, $x\ne 0$, such that

$$x \wedge u=0\mathrm{for}\mathrm{all}u\in I_{+}.$$

Consider the constant net $\left(x\right)$, i.e., $x_{\alpha }=x$ for all $\alpha$. For any $u\in I_{+}$, we have

$$|x_{\alpha }-0| \wedge u=x \wedge u=0,$$

so that $x_{\alpha }\stackrel{u_I\lambda }{\to }0$. But, also, trivially,

$$|x_{\alpha }-x| \wedge u=0\mathrm{for}\mathrm{all}u\in I_{+},$$

so $x_{\alpha }\stackrel{u_I\lambda }{\to }x$. Hence, the net $\left(x\right)$ converges to both 0 and x, contradicting the Hausdorff property. Therefore, I must be order-dense. For further clarification, we refer the reader to standard results on order density in vector lattices (see, e.g., [17]).

(b) Sufficiency. Conversely, assume that I is order-dense and that $\lambda$ is Hausdorff. Let $\left(x_{\alpha }\right)$ be a net in X such that

$$x_{\alpha }\stackrel{u_I\lambda }{\to }x\mathrm{and}{x}_{\alpha }\stackrel{u_I\lambda }{\to }y.$$

Then, for every $u\in I_{+}$,

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathrm{and}|x_{\alpha }-y| \wedge u\stackrel{\lambda }{\to }0.$$

By the triangle inequality,

$$|x-y| \wedge u \le |x_{\alpha }-x| \wedge u + |x_{\alpha }-y| \wedge u.$$

Taking limits (with respect to $\lambda$) on both sides gives

$$|x-y| \wedge u=0\mathrm{for}\mathrm{all}u\in I_{+}.$$

Since I is order-dense, this forces $|x-y|=0$, and, hence, $x=y$. Therefore, $u_I\lambda$ is Hausdorff.

(3) Let $\left(x_{\alpha }\right)$ be a $u_I\lambda$-Cauchy net in X, meaning that for every $u\in I_{+}$ and every $ϵ>0$, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$|x_{\alpha }-x_{\beta }| \wedge u\stackrel{\lambda }{\to }0.$$

Fix $u\in I_{+}$. Since $\lambda$ is a locally solid convergence structure, the net $\left(|x_{\alpha }-x_{\beta }| \wedge u\right)$ is Cauchy in $\lambda$. By the order-completeness of X, every order Cauchy net has an order limit. Thus, there exists an element $x\in X$ such that for every $u\in I_{+}$,

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0,$$

i.e., $x_{\alpha }\stackrel{u_I\lambda }{\to }x$. Hence, every $u_I\lambda$-Cauchy net converges in X, showing that $u_I\lambda$ preserves completeness.  □

Remark 2 

(On Cauchy Nets and Order Completeness). In Proposition 1 (Part (3)), we assert that every $u_I\lambda$-Cauchy net has an order limit in X, relying on the order-completeness of the vector lattice X. To clarify this step, we note the following standard result (see, e.g., [18], [Theorem 2.4]): in an order-complete vector lattice, every order Cauchy net converges in order.

More precisely, let $\left(x_{\alpha }\right)$ be a $u_I\lambda$-Cauchy net, meaning that for every $u\in I^{+}$ and every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$|x_{\alpha }-x_{\beta }| \wedge u\stackrel{\lambda }{\to }0.$$

Since the net $(|x_{\alpha }-x_{\beta }| \wedge u)$ is order-bounded by u, the order-completeness of X ensures that $\left(x_{\alpha }\right)$ is order-convergent to some $x\in X$. Furthermore, by the absorption property of solid neighborhoods in the locally solid convergence structure, this order convergence implies convergence in the $u_I\lambda$-sense.

Thus, the transition from a net being Cauchy to the existence of a limit element in X is well justified under the assumption of order-completeness.

Remark 3 

(Counterexample Illustrating the Necessity of Hausdorffness). Consider the vector lattice $X=\mathbb{R}$ equipped with the indiscrete convergence structure λ, defined by declaring that every net in X converges to every point in X. That is, for any net $\left(x_{\alpha }\right)$ and for every $x\in \mathbb{R}$, we have

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

In this setting, the convergence structure is clearly non-Hausdorff since the uniqueness of limits fails dramatically: for any constant net (for example, $\left(1\right)$), we have

$$1\stackrel{\lambda }{\to }0\mathit{and}1\stackrel{\lambda }{\to }1.$$

This counterexample demonstrates that without the Hausdorff assumption, the convergence structure becomes ill-behaved, as a single net may converge to multiple, distinct points.

Moreover, when working within the framework of unbounded convergence using an ideal I, if I is not order-dense, a similar phenomenon occurs. Specifically, there exists a nonzero $x\in X_{+}$ such that

$$x \wedge u=0\mathit{for}\mathit{all}u\in I^{+},$$

so that the constant net $\left(x\right)$ converges to both 0 and x, contradicting the Hausdorff property.

Remark 4 

(Uniqueness of Limits in the Order Dense Case). In the situation where the ideal I is order-dense in X, the locally solid convergence structure ensures that limits are unique. To elaborate, suppose that a net $\left(x_{\alpha }\right)$ converges to two points x and y in the $u_I\lambda$-convergence structure. Then, for every $u\in I^{+}$, we have

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathit{and}|x_{\alpha }-y| \wedge u\stackrel{\lambda }{\to }0.$$

By applying the triangle inequality and using the order density of I, one can deduce that

$$|x-y| \wedge u\stackrel{\lambda }{\to }0\mathit{for}\mathit{every}u\in I^{+}.$$

Since I is order-dense, the condition $|x-y| \wedge u=0$ for all $u\in I^{+}$ forces $|x-y|=0$, thereby implying $x=y$. This result is standard in the theory of locally solid vector lattices (see, e.g., [18], [Theorem 2.4]).

Thus, the order density of I guarantees the uniqueness of limits in the $u_I\lambda$-convergence structure.

Theorem 3 

(Minimality of $u_{\mathcal{I}}\lambda$). Let λ be a locally solid convergence and $\mathcal{I}$ an ideal in X. Then, $u_{\mathcal{I}}\lambda$ is the weakest locally solid convergence structure that agrees with λ on order-bounded sets.

Proof. 

Assume that

$$x_{\alpha }\stackrel{u_I\lambda }{\to }x.$$

By definition, for every $u\in I_{+}$, we have

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0.$$

Since the net $(|x_{\alpha }-x| \wedge u)$ is order-bounded (indeed, it is dominated by u) and since $\mu$ agrees with $\lambda$ on order-bounded sets, it follows that

$$|x_{\alpha }-x| \wedge u\stackrel{\mu }{\to }0\mathrm{for}\mathrm{every}u\in I_{+}.$$

Let V be an arbitrary $\mu$-neighborhood of 0. Since $\mu$ is locally solid, there exists a solid $\mu$-neighborhood W with

$$W\subset V.$$

By the absorption property of solid neighborhoods, there exists an element $u_0\in I_{+}$ such that for any $y\in X$, the condition

$$|y| \wedge u_0\in W$$

implies $y\in W$. (This is possible because in a locally solid convergence structure the solid neighborhoods serve as absorbing sets).

Since

$$|x_{\alpha }-x| \wedge u_0\stackrel{\mu }{\to }0,$$

there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$, we have

$$|x_{\alpha }-x| \wedge u_0\in W.$$

By the choice of $u_0$ and the solidity of W, this implies that for all $\alpha \ge {\alpha }_0$,

$$x_{\alpha }-x\in W\subset V.$$

Thus, $x_{\alpha }\stackrel{\mu }{\to }x$.

Since V was an arbitrary $\mu$-neighborhood of 0, we conclude that every net converging to x in $u_I\lambda$ also converges to x in $\mu$. In other words, $u_I\lambda$ is weaker than (or equal to) any locally solid convergence structure $\mu$ that agrees with $\lambda$ on order-bounded sets. This proves the minimality of $u_I\lambda$.  □

If $\mathcal{I}=X$, then $u_{\mathcal{I}}\lambda$ coincides with the classical unbounded modification $u\lambda$. If $\mathcal{I}$ is a band, then the structure adapts convergence selectively within the band structure.

Remark 5 

(The Choice of the Solid y-Neighborhood W). In our proof, the solid neighborhood W is chosen to satisfy two key properties: (i) it is absorbing and (ii) it allows for control of the modulus operation such that if $|y| \wedge u_0\in W$, then $y\in W$. This choice is standard in the context of locally solid convergence structures and is essential for transferring the convergence of $|x_{\alpha }-x| \wedge u_0$ into the convergence of $x_{\alpha }-x$ itself.

Alternative selections of W could be made; however, they would still need to satisfy the same absorption and solidity conditions. In our setting, the current selection is optimal because it directly leverages the established absorption property of solid neighborhoods (see Lemma on the Absorption Property in Section 2) and yields a clear and straightforward bound in the subsequent arguments. We note that other choices that satisfy these properties would lead to similar conclusions, and, hence, the current choice does not impose any additional restrictions but rather simplifies the overall presentation.

Definition 7 

(Operator Unbounded Convergence). Let $T:X\to Y$ be an order-bounded operator between vector lattices. We say that a net $\left(x_{\alpha }\right)$ in X satisfies operator-unbounded convergence, denoted $x_{\alpha }\stackrel{u_T\lambda }{\to }x$, if

$$T(x_{\alpha }-x)\stackrel{\lambda }{\to }0.$$

Theorem 4 

(Characterization of Operator-Unbounded Convergence). Let $T:X\to Y$ be a positive operator between Banach lattices. Then, $x_{\alpha }\stackrel{u_T\lambda }{\to }x$ if and only if $|T(x_{\alpha }-x)| \wedge u\stackrel{\lambda }{\to }0$ for all $u\in Y_{+}$.

Proof. 

We prove the necessity and sufficiency as the following:

Necessity:

Suppose that

$$T(x_{\alpha }-x)\stackrel{\lambda }{\to }0.$$

Since $\lambda$ is a locally solid convergence structure, it has the property that if a net $\left(y_{\alpha }\right)$ converges to zero, then for every $u\in Y_{+}$, the net $(|y_{\alpha }| \wedge u)$ also converges to zero. Applying this to

$$y_{\alpha }:=T(x_{\alpha }-x),$$

we deduce that for every $u\in Y_{+}$,

$$|T(x_{\alpha }-x)| \wedge u\stackrel{\lambda }{\to }0.$$

Sufficiency:

Conversely, assume that for every $u\in Y_{+}$,

$$|T(x_{\alpha }-x)| \wedge u\stackrel{\lambda }{\to }0.$$

We wish to show that $T(x_{\alpha }-x)\stackrel{\lambda }{\to }0$. Let V be an arbitrary $\lambda$-neighborhood of 0 in Y. By the local solidity of $\lambda$, there exists a solid neighborhood U such that

$$U\subset V.$$

Since U is absorbing and solid, there exists an element $u_0\in Y_{+}$ satisfying the following: for every $y\in Y$, if

$$|y| \wedge u_0\in U,$$

then $y\in U$.

By our assumption,

$$|T(x_{\alpha }-x)| \wedge u_0\stackrel{\lambda }{\to }0,$$

so there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$,

$$|T(x_{\alpha }-x)| \wedge u_0\in U.$$

Since U is solid, it follows that for all $\alpha \ge {\alpha }_0$,

$$T(x_{\alpha }-x)\in U\subset V.$$

Because V was an arbitrary $\lambda$-neighborhood of 0, we conclude that

$$T(x_{\alpha }-x)\stackrel{\lambda }{\to }0.$$

We thus show that

$$x_{\alpha }\stackrel{u_T\lambda }{\to }x⟺|T(x_{\alpha }-x)| \wedge u\stackrel{\lambda }{\to }0\mathrm{for}\mathrm{every}u\in Y_{+},$$

which completes the proof.  □

This result extends the classical unbounded norm convergence in Banach lattices to a broader class of operator-induced convergence structures.

Remark 6 

(The Necessity of the Positivity Assumption in Theorem 4). In Theorem 4, the positivity of the operator T is crucial because it ensures that T preserves the order structure and modulus of elements in the Banach lattice Y. Specifically, if T is positive, then for any net $\left(x_{\alpha }\right)$ in X converging to x, we have

$$|T(x_{\alpha }-x)| \wedge u \le T(|x_{\alpha }-x|) \wedge u,$$

and since T preserves the modulus, it follows that the convergence

$$|T(x_{\alpha }-x)| \wedge u\stackrel{\lambda }{\to }0\mathit{for}\mathit{every}u\in Y_{+}$$

directly follows from the unbounded convergence of $\left(x_{\alpha }\right)$ in X. This property is essential to deduce the operator-unbounded convergence, and, without the positivity assumption, T may not preserve the necessary order properties, and the above inequality might fail.

Moreover, if T were not positive, counterexamples could be constructed where the preservation of the order structure would not be lost, leading to failure of the desired convergence properties. For example, a non-positive operator might reverse the order of some elements, causing the modulus operation to behave unpredictably under T, thereby invalidating the step where we conclude that

$$|T(x_{\alpha }-x)| \wedge u\to 0.$$

Thus, the positivity assumption in Theorem 3 is not only standard but also essential for the argument, and any extension of the result to a broader class of operators would require additional conditions to compensate for the loss of order preservation.

4. Bornological and Unbounded Convergences

Definition 8 

(Bornology [9]). Let X be a vector lattice. A bornology $\mathcal{B}$ on X is a collection of subsets of X satisfying the following:

1. 

$\mathcal{B}$ covers X, i.e., for every $x\in X$, there exists $B\in \mathcal{B}$ such that $x\in B$.

2. 

$\mathcal{B}$ is upward hereditary: if $B\in \mathcal{B}$ and $A\subseteq B$, then $A\in \mathcal{B}$.

3. 

$\mathcal{B}$ is stable under finite unions.

Elements of $\mathcal{B}$ are called bounded sets.

Definition 9 

(Bounded Modification of a Convergence Structure [10]). Let λ be a locally solid convergence structure on X and let $\mathcal{B}$ be a bornology on X. The bounded modification of λ, denoted $b\lambda$, is defined thus: a net $\left(x_{\alpha }\right)$ converges to x in $b\lambda$, written as

$$x_{\alpha }\stackrel{b\lambda }{\to }x,$$

if there exists a bounded set $B\in \mathcal{B}$ such that, eventually, $x_{\alpha }-x\in B$ and $\left(x_{\alpha }\right)$ converges to x in λ.

Proposition 2. 

Let λ be a locally solid convergence structure on X and $\mathcal{B}$ a bornology on X. Then, the bounded modification $b\lambda$ is a locally solid convergence structure that coincides with λ on bounded sets.

Proof. 

Since $\lambda$ is locally solid, the ordering structure and solidity of neighborhoods are preserved on any bounded set. The bornology $\mathcal{B}$ guarantees that the convergence behavior is controlled on these sets. Hence, on any bounded set $B\in \mathcal{B}$, the convergence in $b\lambda$ is identical to that in $\lambda$, while, on unbounded sets, the convergence is localized through the condition $x_{\alpha }-x\in B$ for some $B\in \mathcal{B}$ (see [9,10]).  □

Definition 10 

(Bornological Unbounded Convergence). Let λ be a locally solid convergence structure on X and $\mathcal{B}$ a bornology on X. The bornological unbounded modification $u_{\mathcal{B}}\lambda$ is defined by

$$x_{\alpha }\stackrel{u_{\mathcal{B}}\lambda }{\to }x\mathit{if}\mathit{and}\mathit{only}\mathit{if}|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathit{for}\mathit{all}u\in {\mathcal{B}}_{+},$$

where ${\mathcal{B}}_{+}=\{u\in X_{+}:u\in B\mathrm{for}\mathrm{some}B\in \mathcal{B}\}$.

Theorem 5 

(Equivalence on Bounded Sets). Let λ be a locally solid convergence structure on X and let $\mathcal{B}$ be a saturated bornology (i.e., every order-bounded set belongs to $\mathcal{B}$). Then, the bornological unbounded modification $u_{\mathcal{B}}\lambda$ coincides with the classical unbounded modification $u\lambda$ on bounded sets.

Proof. 

Let $\lambda$ be a locally solid convergence structure on a vector lattice X and let $\mathcal{B}$ be a saturated bornology on X, meaning that every order-bounded subset of X belongs to $\mathcal{B}$. By definition, the classical unbounded modification $u\lambda$ is given by

$$x_{\alpha }\stackrel{u\lambda }{\to }x⟺|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathrm{for}\mathrm{all}u\in X_{+},$$

while the bornological unbounded modification $u_B\lambda$ is defined by

$$x_{\alpha }\stackrel{u_B\lambda }{\to }x⟺|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathrm{for}\mathrm{all}u\in B_{+},$$

where

$$B_{+}=\{u\in X_{+}:u\in B\mathrm{for}\mathrm{some}B\in \mathcal{B}\}.$$

Since $\mathcal{B}$ is saturated, every order-bounded subset of X belongs to $\mathcal{B}$. In particular, for any $u\in X_{+}$, the singleton $\left\{u\right\}$ is order-bounded, and, hence, $u\in B$ for some $B\in \mathcal{B}$. This shows that

$$X_{+}\subseteq B_{+}.$$

On the other hand, by definition, we always have $B_{+}\subseteq X_{+}$. Therefore,

$$B_{+}=X_{+}.$$

It follows immediately that for any net $\left(x_{\alpha }\right)$ in X,

$$x_{\alpha }\stackrel{u_B\lambda }{\to }x⟺x_{\alpha }\stackrel{u\lambda }{\to }x.$$

Thus, on bounded sets (where the saturation condition applies), the bornological unbounded modification $u_B\lambda$ coincides with the classical unbounded modification $u\lambda$.  □

Lemma 6 

(Uniform Continuity on Bounded Sets). Let λ be a locally solid convergence structure on X and let $\left(x_{\alpha }\right)$ be a net eventually contained in a bounded set $B\in \mathcal{B}$. Then, if $x_{\alpha }\stackrel{u\lambda }{\to }x$, it follows that $x_{\alpha }\stackrel{b\lambda }{\to }x$.

Proof. 

Let $\left(x_{\alpha }\right)$ be a net in X and suppose there exists a bounded set $B\in \mathcal{B}$ and an index ${\alpha }_0$ such that

$$x_{\alpha }\in B\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_0.$$

Assume that

$$x_{\alpha }\stackrel{u\lambda }{\to }x.$$

By definition of the unbounded modification, for every $u\in X_{+}$, we have

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0.$$

Since $\left(x_{\alpha }\right)$ is eventually contained in B, the differences

$$x_{\alpha }-x(\alpha \ge {\alpha }_0)$$

are eventually contained in the translate $B-x$, which is also a bounded set.

On bounded sets, the lattice operations (in particular, the modulus) are uniformly continuous with respect to the locally solid convergence structure $\lambda$. In other words, for every $\lambda$-neighborhood U of 0, there exists a $\lambda$-neighborhood V of 0 such that for all $y\in B-x$,

$$|y|\in V⟹y\in U.$$

Applying this uniform continuity to the net $(x_{\alpha }-x)$, the fact that

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathrm{for}\mathrm{all}u\in X_{+}$$

implies that for every $\lambda$-neighborhood U of 0, there exists an index ${\alpha }_1\ge {\alpha }_0$ such that

$$x_{\alpha }-x\in U\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_1.$$

This is precisely the definition of convergence in the bounded modification $b\lambda$. Hence,

$$x_{\alpha }\stackrel{b\lambda }{\to }x.$$

 □

The bounded modification $b\lambda$ acts as a bridge between classical topological convergence and unbounded convergence. While $u\lambda$ extends convergence globally, $b\lambda$ localizes convergence to bounded subsets, providing finer control in applications where boundedness is inherent (e.g., in function spaces or operator theory) [20].

Definition 11 

(Bornological Dual Convergence). Let $(X,\lambda )$ be a locally solid convergence space and $\mathcal{B}$ a bornology on X. The bornological dual convergence on the dual space $X^{*}$ is defined by declaring that a net $\left(f_{\alpha }\right)$ in $X^{*}$ converges to $f\in X^{*}$ if for every $B\in \mathcal{B}$ the net ${\left(f_{\alpha }\left(x\right)\right)}_{x\in B}$ converges uniformly to ${\left(f\left(x\right)\right)}_{x\in B}$.

Proposition 3 

(Bounded Duality). Let X be a vector lattice with a locally solid convergence structure λ and a saturated bornology $\mathcal{B}$. Then, the dual convergence induced on $X^{*}$ by $b\lambda$ is equivalent to the dual topology induced by uniform convergence on bounded sets.

Proof. 

By definition, a net $\left(f_{\alpha }\right)$ in $X^{*}$ converges to $f\in X^{*}$ with respect to the dual convergence induced by $b\lambda$ if for every bounded set $B\in \mathcal{B}$, the following holds:

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|\to 0\mathrm{as}\alpha \to \infty .$$

In other words, the convergence is uniform on every bounded set.

The dual topology on $X^{*}$, induced by uniform convergence on bounded sets, is defined via the family of seminorms:

$$p_B\left(g\right)=\underset{x\in B}{sup}\left|g\left(x\right)\right|,B\in \mathcal{B}.$$

A net $\left(f_{\alpha }\right)$ converges to f in this topology if and only if for every $B\in \mathcal{B}$ and every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$,

$$p_B(f_{\alpha }-f)=\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\epsilon .$$

Since the bounded modification $b\lambda$ agrees with $\lambda$ on bounded sets (by its very definition), the uniform convergence on bounded sets with respect to $b\lambda$ is identical to the uniform convergence on bounded sets with respect to $\lambda$.

Thus, if $\left(f_{\alpha }\right)$ converges to f in the dual convergence induced by $b\lambda$, then for every $B\in \mathcal{B}$ and $\epsilon >0$, there exists an index ${\alpha }_0$ such that

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\epsilon \mathrm{for}\mathrm{all}\alpha \ge {\alpha }_0.$$

This is precisely the condition for convergence in the dual topology defined by the seminorms $p_B$. Conversely, if $\left(f_{\alpha }\right)$ converges uniformly on every bounded set (i.e., in the dual topology), then, by definition, it converges with respect to the dual convergence induced by $b\lambda$.

We have shown that the dual convergence induced on $X^{*}$ by $b\lambda$ is equivalent to the dual topology defined by uniform convergence on bounded sets. In other words, the two notions coincide.  □

The integration of bornological modifications into the unbounded convergence framework enriches the classical theory of locally solid convergence structures. It allows one to capture subtle behaviors on bounded subsets, which is particularly useful in applications to function spaces and operator theory. Moreover, this approach opens promising avenues for further research in duality theory and ergodic convergence [5,20].

5. Locally Solid Convergence in Functional Analysis

Definition 12 

(Continuous Convergence on $C(X,Y)$ [5,9]). Let X be a convergence space and Y be a locally solid convergence space. A net $\left(f_{\alpha }\right)$ in $C(X,Y)$ (the space of continuous functions from X to Y) is said to converge continuously to $f\in C(X,Y)$, denoted as

$$f_{\alpha }\stackrel{c}{\to }f,$$

if for every $x\in X$ and every net $\left(x_{\beta }\right)$ in X satisfying $x_{\beta }\to x$, the double net $\left(f_{\alpha }\left(x_{\beta }\right)\right)$ converges to $f\left(x\right)$ in Y.

Proposition 4. 

Let X be a convergence space and Y be a locally solid convergence space. Then, the continuous convergence on $C(X,Y)$ defines a locally solid convergence structure on $C(X,Y)$.

Proof. 

For every $x\in X$ and every net $\left(x_{\beta }\right)$ converging to x, the local solidity of Y guarantees that if $f_{\alpha }\left(x\right)$ converges to $f\left(x\right)$ and $|g_{\alpha }\left(x\right)| \le |f_{\alpha }\left(x\right)|$, then $g_{\alpha }\left(x\right)$ also converges to $f\left(x\right)$. Uniformity over appropriate subsets of X (e.g., compact or bounded subsets) preserves the lattice operations, ensuring that the induced convergence structure on $C(X,Y)$ is locally solid.  □

Definition 13 

(Relative Uniform Convergence on $C(X,Y)$). Let Y be a vector lattice. A net $\left(f_{\alpha }\right)$ in $C(X,Y)$ is said to converge relatively uniformly to $f\in C(X,Y)$, denoted as

$$f_{\alpha }\stackrel{ru}{\to }f,$$

if there exists a regulator function $e\in C{(X,Y)}_{+}$ such that for every $\epsilon >0$, there exists an index ${\alpha }_0$ satisfying

$$|f_{\alpha }\left(x\right)-f\left(x\right)| \le \epsilon e\left(x\right)\mathit{for}\mathit{all}\alpha \ge {\alpha }_0\mathit{and}\mathit{all}x\in X.$$

Theorem 6. 

Assume that Y is an order-continuous Banach lattice. Then, relative uniform convergence on $C(X,Y)$ implies continuous convergence, i.e., if $f_{\alpha }\stackrel{ru}{\to }f$, then $f_{\alpha }\stackrel{c}{\to }f$.

Proof. 

For each $x\in X$, the inequality

$$|f_{\alpha }\left(x\right)-f\left(x\right)| \le \epsilon e\left(x\right)$$

ensures that $f_{\alpha }\left(x\right)$ converges to $f\left(x\right)$ in the order topology of Y. Since Y is order-continuous, order convergence implies convergence in the locally solid structure of Y. Uniform control provided by e yields the continuous convergence of the net $\left(f_{\alpha }\right)$ to f on X.  □

Definition 14 

(Operator-Induced Convergence). Let $T:C(X,Y)\to Z$ be a linear operator, where Z is a vector lattice with a locally solid convergence structure. We say that T is convergence-preserving if for every net $\left(f_{\alpha }\right)$ in $C(X,Y)$ converging continuously to f, we have

$$T\left(f_{\alpha }\right)\stackrel{\lambda }{\to }T\left(f\right)$$

in Z.

Theorem 7 

(Convergence Preservation by Positive Operators). Let $T:C(X,Y)\to Z$ be a positive linear operator, where Y and Z are locally solid convergence vector lattices. If $\left(f_{\alpha }\right)$ converges continuously to f in $C(X,Y)$, then

$$T\left(f_{\alpha }\right)\stackrel{c}{\to }T\left(f\right)$$

in Z.

Proof. 

Let $T:C(X,Y)\to Z$ be a positive linear operator, where Y and Z are locally solid convergence vector lattices. Assume that

$$f_{\alpha }\stackrel{c}{\to }f\mathrm{in}C(X,Y),$$

meaning that for every $x\in X$ and every net $\left(x_{\beta }\right)$ in X with

$$x_{\beta }\to x,$$

the double net

$$\left(f_{\alpha }\left(x_{\beta }\right)\right)$$

converges in Y to $f\left(x\right)$.

We need to show that

$$T\left(f_{\alpha }\right)\stackrel{c}{\to }T\left(f\right)\mathrm{in}Z.$$

For each $x\in X$, by the linearity and positivity of T, we have

$$\left|T\left(f_{\alpha }\right)\left(x\right)-T\left(f\right)\left(x\right)\right|=\left|T(f_{\alpha }-f)\left(x\right)\right| \le T\left(|f_{\alpha }-f|\right)\left(x\right).$$

(1)

Since T is positive, the above inequality holds in the order of Z.

Because $\left(f_{\alpha }\right)$ converges continuously to f in $C(X,Y)$, for each fixed $x\in X$, we have

$$|f_{\alpha }\left(x\right)-f\left(x\right)|\stackrel{{\lambda }_Y}{\to }0,$$

where ${\lambda }_Y$ denotes the locally solid convergence in Y. By the solidity of the locally solid structure, this implies that for every regulator $u\in Y_{+}$,

$$|f_{\alpha }\left(x\right)-f\left(x\right)| \wedge u\stackrel{{\lambda }_Y}{\to }0.$$

Since T is continuous with respect to the locally solid convergence (and positive), we obtain

$$T\left(|f_{\alpha }-f|\right)\left(x\right)\stackrel{{\lambda }_Z}{\to }0\mathrm{for}\mathrm{every}x\in X,$$

where ${\lambda }_Z$ is the locally solid convergence in Z. Hence,

$$\left|T\left(f_{\alpha }\right)\left(x\right)-T\left(f\right)\left(x\right)\right|\stackrel{{\lambda }_Z}{\to }0\mathrm{for}\mathrm{each}x\in X.$$

Continuous convergence on $C(X,Y)$ requires not only point-wise convergence but also that the convergence is uniform on subsets of X relevant to the topology (or bornology) of X. Since T is linear and positive, and because the convergence of $\left(f_{\alpha }\right)$ to f is continuous (hence, uniform on the appropriate subsets), the estimate in (1) transfers this uniformity to the net $\left(T\left(f_{\alpha }\right)\right)$. In other words, for every $x\in X$ and every net $\left(x_{\beta }\right)$ with $x_{\beta }\to x$, the double net

$$T\left(f_{\alpha }\right)\left(x_{\beta }\right)$$

converges in Z to $T\left(f\right)\left(x\right)$.

Thus, we have shown that for every $x\in X$, $T\left(f_{\alpha }\right)\left(x\right)$ converges to $T\left(f\right)\left(x\right)$ in Z, and the convergence is uniform on the appropriate subsets of X. Therefore,

$$T\left(f_{\alpha }\right)\stackrel{c}{\to }T\left(f\right)\mathrm{in}Z.$$

This completes the proof.  □

Proposition 5 

(Duality for $C(X,Y)$). Let X be a compact space and Y be an order-continuous Banach lattice with a locally solid convergence structure. Then, the dual space $C{(X,Y)}^{*}$, equipped with the topology of uniform convergence on bounded sets, can be identified with the space of regular Borel measures with values in $Y^{*}$.

Proof. 

Let X be a compact Hausdorff space and let Y be an order-continuous Banach lattice endowed with a locally solid convergence structure. Consider the Banach lattice $C(X,Y)$ of continuous functions from X into Y, equipped with the topology of uniform convergence on bounded sets. We wish to show that every continuous linear functional

$$L:C(X,Y)\to \mathbb{R}\left(\mathrm{or}\mathbb{C}\right)$$

can be represented uniquely in the form

$$L\left(f\right)={\int }_{X}fd\mu ,\mathrm{for}\mathrm{all}f\in C(X,Y),$$

where $\mu$ is a regular Borel measure on X with values in $Y^{*}$ and the integral is defined in the vector-valued sense.

For each $y^{*}\in Y^{*}$, define the scalar linear functional

$$L_{y^{*}}:C\left(X\right)\to \mathbb{R}\mathrm{by}{L}_{y^{*}}\left(g\right)=y^{*}\left(L\left(f\right)\right),$$

where $f\in C(X,Y)$ is any function satisfying

$$g\left(x\right)=y^{*}\left(f\left(x\right)\right)\mathrm{for}\mathrm{all}x\in X.$$

Since the evaluation maps $f\mapsto y^{*}\left(f\left(x\right)\right)$ are continuous and L is continuous on $C(X,Y)$, it follows that $L_{y^{*}}$ is a continuous linear functional on $C\left(X\right)$. By the classical Riesz representation theorem, there exists a unique regular Borel measure ${\mu }_{y^{*}}$ on X such that

$$L_{y^{*}}\left(g\right)={\int }_{X}gd{\mu }_{y^{*}}\mathrm{for}\mathrm{all}g\in C\left(X\right).$$

We now show that the family ${\left\{{\mu }_{y^{*}}\right\}}_{y^{*}\in Y^{*}}$ arises from a unique vector measure

$$\mu :\mathcal{B}\left(X\right)\to Y^{*}$$

with the property that for every Borel set $E\subset X$ and every $y^{*}\in Y^{*}$,

$$y^{*}\left(\mu \left(E\right)\right)={\mu }_{y^{*}}\left(E\right).$$

Indeed, for each fixed Borel set $E\subset X$, the mapping

$$Y^{*}\to \mathbb{R},y^{*}\mapsto {\mu }_{y^{*}}\left(E\right)$$

is linear. Moreover, the order continuity of Y guarantees that this mapping is continuous (with respect to the weak* topology on $Y^{*}$). Hence, by the Hahn–Banach theorem, there exists a unique element $\mu \left(E\right)\in Y^{*}$ such that

$$y^{*}\left(\mu \left(E\right)\right)={\mu }_{y^{*}}\left(E\right)\mathrm{for}\mathrm{all}{y}^{*}\in Y^{*}.$$

Standard arguments then show that $\mu$ is a regular Borel measure with values in $Y^{*}$.

For any $f\in C(X,Y)$ and any $y^{*}\in Y^{*}$, we can define the scalar function

$$g_{y^{*}}\left(x\right)=y^{*}\left(f\left(x\right)\right),x\in X.$$

Then, by the definition of $L_{y^{*}}$ and the construction of $\mu$, we have

$$y^{*}\left(L\left(f\right)\right)=L_{y^{*}}\left(g_{y^{*}}\right)={\int }_X{g}_{y^{*}}d{\mu }_{y^{*}}={\int }_X{y}^{*}\left(f\left(x\right)\right)d{\mu }_{y^{*}}\left(x\right)=y^{*}\left({\int }_{X}f\left(x\right)d\mu \left(x\right)\right).$$

Since this equality holds for every $y^{*}\in Y^{*}$, by the Hahn–Banach theorem, it follows that

$$L\left(f\right)={\int }_{X}fd\mu .$$

The topology on $C{(X,Y)}^{*}$ induced by uniform convergence on bounded sets is given by the family of seminorms

$$p_B\left(L\right)=\underset{f\in B}{sup}|L\left(f\right)|,B\in \mathcal{B}\left(C(X,Y)\right).$$

Standard arguments (see, e.g., [22]) show that the mapping

$$L\mapsto \mu$$

defines an isometric isomorphism between $C{(X,Y)}^{*}$ and the space of regular Borel measures with values in $Y^{*}$.

We have thus established that every continuous linear functional on $C(X,Y)$ can be represented uniquely as

$$L\left(f\right)={\int }_{X}fd\mu ,$$

with $\mu$ being a regular Borel measure with values in $Y^{*}$. The dual space $C{(X,Y)}^{*}$, equipped with the topology of uniform convergence on bounded sets, is isometrically isomorphic to the space of such vector measures. This completes the proof.  □

The interplay between continuous convergence and relative uniform convergence in $C(X,Y)$ provides a rich framework for studying duality and operator adjoints. In particular, the locally solid framework allows us to analyze reflexivity properties and further explore the structure of the dual space.

6. Duality and Reflexivity

In this section, we deepen the investigation of dual spaces associated with locally solid convergence structures. We introduce novel concepts and results that provide insights into reflexivity, spectral duality, and ergodic properties in this setting.

Definition 15 

(Dual Space and Dual Convergence). Let $(X,\lambda )$ be a locally solid convergence space (in particular, a locally solid convergence vector lattice). The dual space $X^{*}$ is defined as the collection of all λ-continuous linear functionals $f:X\to \mathbb{R}$. A net $\left(f_{\alpha }\right)$ in $X^{*}$ is said to converge in the dual convergence structure (or ${\lambda }^{*}$-converge) to $f\in X^{*}$ if, for every $x\in X$ (or, more generally, uniformly on each bounded subset of X), we have

$$f_{\alpha }\left(x\right)\to f\left(x\right)$$

with respect to the convergence induced by the usual topology on $\mathbb{R}$.

Remark 7. 

In many applications, one equips $X^{*}$ with the topology of uniform convergence on bounded sets (or on the λ-bounded subsets of X), which is naturally compatible with the locally solid structure on X (see, e.g., [1,9]).

Definition 16 

(Reflexivity). A locally solid convergence vector lattice X is said to be reflexive if the natural embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)\mathit{for}\mathit{all}f\in X^{*}$$

is surjective. In this case, every ${\lambda }^{*}$-continuous linear functional on $X^{*}$ arises from evaluation at some $x\in X$.

Lemma 7 

(Continuity of the Evaluation Mapping). Let X be a locally solid convergence vector lattice and $X^{*}$ be its dual space with the dual convergence structure. Then, the evaluation mapping

$$E:X\times X^{*}\to \mathbb{R},E(x,f)=f\left(x\right)$$

is jointly continuous in the sense that if $x_{\alpha }\stackrel{\lambda }{\to }x$ in X and $f_{\beta }\stackrel{{\lambda }^{*}}{\to }f$ in $X^{*}$, then

$$f_{\beta }\left(x_{\alpha }\right)\to f\left(x\right),$$

provided that the convergence is uniform on bounded sets.

Proof. 

Let $\epsilon >0$ be arbitrary. Since $f\in X^{*}$ is $\lambda$-continuous and $x_{\alpha }\stackrel{\lambda }{\to }x$ in X, by the continuity of f, there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$,

$$\left|f\left(x_{\alpha }\right)-f\left(x\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

On the other hand, the dual convergence structure on $X^{*}$ is defined via uniform convergence on bounded sets. In particular, since the net $\left(f_{\beta }\right)$ converges to f in $X^{*}$ uniformly on bounded subsets of X, there exists a bounded set $B\subseteq X$ and an index ${\beta }_0$ such that for all $\beta \ge {\beta }_0$,

$$\underset{y\in B}{sup}\left|f_{\beta }\left(y\right)-f\left(y\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

Moreover, because $x_{\alpha }\stackrel{\lambda }{\to }x$, the net $\left(x_{\alpha }\right)$ is eventually contained in any neighborhood of x, so, we may assume, by choosing a larger index if necessary, that for all $\alpha \ge {\alpha }_1\ge {\alpha }_0$, we have

$$x_{\alpha }\in B.$$

Then, for all $\alpha \ge {\alpha }_1$ and $\beta \ge {\beta }_0$, the triangle inequality yields

$$\left|f_{\beta }\left(x_{\alpha }\right)-f\left(x\right)\right| \le \left|f_{\beta }\left(x_{\alpha }\right)-f\left(x_{\alpha }\right)\right| + \left|f\left(x_{\alpha }\right)-f\left(x\right)\right|.$$

By our choice of ${\beta }_0$, since $x_{\alpha }\in B$, we have

$$\left|f_{\beta }\left(x_{\alpha }\right)-f\left(x_{\alpha }\right)\right|<{\displaystyle \frac{\epsilon }{2}},$$

and by the continuity of f, we have

$$\left|f\left(x_{\alpha }\right)-f\left(x\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

Thus,

$$\left|f_{\beta }\left(x_{\alpha }\right)-f\left(x\right)\right|<{\displaystyle \frac{\epsilon }{2}} + {\displaystyle \frac{\epsilon }{2}}=\epsilon .$$

Since $\epsilon >0$ was arbitrary, this shows that

$$f_{\beta }\left(x_{\alpha }\right)\to f\left(x\right)$$

as $\alpha \to \infty$ and $\beta \to \infty$. Hence, the evaluation mapping

$$E:X\times X^{*}\to \mathbb{R},E(x,f)=f\left(x\right)$$

is jointly continuous.  □

Theorem 8 

(Completeness of the Dual). Let $(X,\lambda )$ be a complete Hausdorff locally solid convergence vector lattice and assume that every bounded subset of X is λ-complete. Then, the dual space $X^{*}$, equipped with the topology of uniform convergence on bounded sets, is complete.

Proof. 

Let $\left(f_{\alpha }\right)$ be a Cauchy net in $X^{*}$ with respect to the topology of uniform convergence on every $\lambda$-bounded subset of X. That is, for every $\lambda$-bounded set $B\subset X$ and every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f_{\beta }\left(x\right)\right|<\epsilon .$$

For each fixed $x\in X$, the singleton $\left\{x\right\}$ is $\lambda$-bounded. Hence, the net $\left(f_{\alpha }\left(x\right)\right)$ is Cauchy in $\mathbb{R}$ (or $\mathbb{C}$) and, by the completeness of the field, converges. We then define

$$f\left(x\right):=\underset{\alpha }{lim}{f}_{\alpha }\left(x\right).$$

This defines the mapping $f:X\to \mathbb{R}$.

Since each $f_{\alpha }$ is linear, for any $x,y\in X$ and scalars $a,b\in \mathbb{R}$, we have

$$f_{\alpha }(ax + by)=a{f}_{\alpha }\left(x\right) + b{f}_{\alpha }\left(y\right).$$

Taking the limit as $\alpha \to \infty$, we obtain

$$f(ax + by)=\underset{\alpha }{lim}{f}_{\alpha }(ax + by)=a\underset{\alpha }{lim}{f}_{\alpha }\left(x\right) + b\underset{\alpha }{lim}{f}_{\alpha }\left(y\right)=af\left(x\right) + bf\left(y\right).$$

Thus, f is linear.

Let $\left(x_{\gamma }\right)$ be a net in X such that $x_{\gamma }\stackrel{\lambda }{\to }x$. Since each $f_{\alpha }$ is $\lambda$-continuous, we have

$$f_{\alpha }\left(x_{\gamma }\right)\to f_{\alpha }\left(x\right)\mathrm{for}\mathrm{each}\mathrm{fixed}\alpha .$$

Moreover, because $\left(f_{\alpha }\right)$ is Cauchy with respect to uniform convergence on bounded sets, for any $\lambda$-bounded set $B\subset X$ (which eventually contains all $x_{\gamma }$) and any $\epsilon >0$, there exists ${\alpha }_0$, such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$\underset{y\in B}{sup}\left|f_{\alpha }\left(y\right)-f_{\beta }\left(y\right)\right|<{\displaystyle \frac{\epsilon }{3}}.$$

Fix $\alpha \ge {\alpha }_0$. Then, by the continuity of $f_{\alpha }$, there exists an index ${\gamma }_0$, such that for all $\gamma \ge {\gamma }_0$,

$$\left|f_{\alpha }\left(x_{\gamma }\right)-f_{\alpha }\left(x\right)\right|<{\displaystyle \frac{\epsilon }{3}}.$$

Taking limits as $\alpha \to \infty$ and using the uniform convergence on B, we obtain for all $\gamma \ge {\gamma }_0$,

$$\left|f\left(x_{\gamma }\right)-f\left(x\right)\right| \le \left|f\left(x_{\gamma }\right)-f_{\alpha }\left(x_{\gamma }\right)\right| + \left|f_{\alpha }\left(x_{\gamma }\right)-f_{\alpha }\left(x\right)\right| + \left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\epsilon .$$

Thus, $x_{\gamma }\stackrel{\lambda }{\to }x$ implies $f\left(x_{\gamma }\right)\to f\left(x\right)$, showing that f is $\lambda$-continuous.

Let $B\subset X$ be any $\lambda$-bounded set and fix $\epsilon >0$. Since $\left(f_{\alpha }\right)$ is Cauchy with respect to the uniform convergence on B, there exists ${\alpha }_0$, such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f_{\beta }\left(x\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

Fix any $\alpha \ge {\alpha }_0$ and let $\beta \to \infty$. Then, by the definition of $f\left(x\right)$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right| \le {\displaystyle \frac{\epsilon }{2}}<\epsilon .$$

This shows that $f_{\alpha }$ converges to f uniformly on every bounded set $B\subset X$.

Since f is linear and $\lambda$-continuous, we have $f\in X^{*}$. Moreover, the net $\left(f_{\alpha }\right)$ converges to f in the topology of uniform convergence on bounded sets. Therefore, the dual space $X^{*}$ is complete.  □

Proposition 6 

(Spectral Characterization in Reflexive Spaces). Let X be a reflexive locally solid convergence vector lattice and let $T:X\to X$ be a positive, λ-continuous operator. Then, the spectrum of T (defined via the duality with $X^{*}$) is nonempty and every spectral value is contained in the closure of the set $\{f\left(Tx\right):x\in X,f\in X^{*},\parallel f\parallel =1\}$.

Proof. 

Since $T:X\to X$ is a positive, $\lambda$-continuous operator on the reflexive locally solid convergence vector lattice X, it is bounded with respect to the topology of uniform convergence on bounded sets. In particular, the spectral radius

$$r\left(T\right)=\underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n}$$

is well defined, and by standard spectral theory (see, e.g., [1]), one has

$$r\left(T\right)\in \sigma \left(T\right).$$

Thus, the spectrum $\sigma \left(T\right)$ is nonempty.

Let $\lambda \in \sigma \left(T\right)$ be an arbitrary spectral value. By the definition of the spectrum, for every $\epsilon >0$, the operator $T-\lambda I$ is not boundedly invertible. Hence, there exists a sequence $\left(x_n\right)$ in X, normalized so that $\parallel x_n\parallel =1$ (with respect to the topology induced by uniform convergence on bounded sets), such that

$$\parallel (T-\lambda I)x_n\parallel <\epsilon \mathrm{for}\mathrm{all}n\mathrm{sufficiently}\mathrm{large}.$$

By the Hahn–Banach theorem and the reflexivity of X, for each n, there exists a functional $f_n\in X^{*}$ with $\parallel f_n\parallel =1$ such that

$$f_n\left(x_n\right)=1.$$

Then, for these $x_n$ and $f_n$, we have

$$|f_n\left(T{x}_n\right)-\lambda | = |f_n\left((T-\lambda I)x_n\right)| \le \parallel (T-\lambda I)x_n\parallel <\epsilon .$$

Since $\epsilon >0$ is arbitrary, it follows that $\lambda$ lies in the closure of the set

$$\{f\left(Tx\right):x\in X,f\in X^{*},\parallel f\parallel =1\}.$$

We have shown that $\sigma \left(T\right)\ne \varnothing$ and that every spectral value $\lambda \in \sigma \left(T\right)$ is contained in the closure of

$$\{f\left(Tx\right):x\in X,f\in X^{*},\parallel f\parallel =1\}.$$

This completes the proof.  □

Remark 8. 

The interplay between the duality of locally solid convergence spaces and spectral theory paves the way for further investigation into ergodic properties. In particular, one may study the convergence of ergodic averages of operators using the dual convergence structure, a topic that remains largely unexplored.

Proposition 7 

(Characterization of Reflexivity). Let X be an order-continuous Banach lattice equipped with a locally solid convergence structure λ such that the norm convergence is contained in λ. Then, X is reflexive if and only if the natural embedding $J:X\to X^{**}$ is λ-dense.

Proof. 

We prove the equivalence in two parts.

(1) If X is reflexive, then $J\left(X\right)$ is $\lambda$-dense in $X^{**}$. By definition, if X is reflexive, then the natural embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*}$$

is surjective, so that $J\left(X\right)=X^{**}$. In particular, $J\left(X\right)$ is dense in $X^{**}$ with respect to any topology (in particular, the $\lambda$-topology) that renders $X^{**}$ Hausdorff.

(2) If $J\left(X\right)$ is $\lambda$-dense in $X^{**}$, then X is reflexive. Assume that $J\left(X\right)$ is $\lambda$-dense in $X^{**}$ and that the dual convergence structure is complete (by Theorem 8). Let $x^{**}\in X^{**}$ be arbitrary. Then, there exists a net $\left(x_{\alpha }\right)$ in X such that

$$J\left(x_{\alpha }\right)\stackrel{\lambda }{\to }{x}^{**}\mathrm{in}{X}^{**}.$$

Since the norm convergence on X is contained in $\lambda$ and X is an order-continuous Banach lattice, the net $\left(J\left(x_{\alpha }\right)\right)$ is Cauchy with respect to the dual convergence structure. By the completeness of the dual, the limit of this net must belong to $X^{**}$. But, by the $\lambda$-density assumption, we have

$$x^{**}\in {\overline{J\left(X\right)}}^{\lambda }=J\left(X\right).$$

Since $x^{**}$ was an arbitrary element of $X^{**}$, it follows that

$$J\left(X\right)=X^{**}.$$

Thus, the natural embedding J is surjective and, hence, X is reflexive.

This completes the proof.  □

The above results indicate that deeper analysis of duality in locally solid convergence spaces can yield new insights into reflexivity. Open questions include the extent to which these duality results can be extended to non-Archimedean or more general ordered spaces and how these concepts interact with ergodic and spectral properties of operators.

Definition 17 

(Dual Locally Solid Convergence Space). Let $(X,\lambda )$ be a locally solid convergence vector lattice. We define the dual space:

$$X^{*}=\{f:X\to \mathbb{R}\left(\mathit{or}\mathbb{C}\right)\mid f\mathit{is}\mathit{linear}\mathit{and}\lambda \text{-}\mathit{continuous}\}.$$

Equip $X^{*}$ with the topology of uniform convergence on bounded sets. That is, a net $\left(f_{\alpha }\right)$ in $X^{*}$ is said to converge in the dual convergence ${\lambda }^{*}$ to $f\in X^{*}$ if for every bounded subset $B\subset X$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|\to 0.$$

The topology on $X^{*}$ induced by uniform convergence on bounded sets preserves a form of local solidity inherited from $\lambda$. This dual convergence structure plays a central role in linking the properties of X with those of its dual.

Definition 18 

(Reflexive Locally Solid Convergence Space). A locally solid convergence vector lattice $(X,\lambda )$ is said to be reflexive if the canonical embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*},$$

is a λ-homeomorphism onto its image and $J\left(X\right)$ is dense in $X^{**}$ with respect to the dual convergence structure ${\lambda }^{*}$.

Lemma 8 

(Uniform Boundedness in the Dual). Let $(X,\lambda )$ be a locally solid convergence vector lattice and let $\left(f_{\alpha }\right)$ be a net in $X^{*}$ converging in ${\lambda }^{*}$. Then, for every bounded set $B\subset X$, there exists a constant $M_B>0$ such that

$$\underset{\alpha }{sup}\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right| \le M_B.$$

Proof. 

Let $\left(f_{\alpha }\right)$ be a net in the dual space $X^{*}$ that converges to some $f\in X^{*}$ with respect to the dual convergence structure ${\lambda }^{*}$, i.e., the topology of uniform convergence on bounded subsets of X. By definition, for every bounded set $B\subset X$, we have

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|⟶0.$$

Fix an arbitrary bounded set $B\subset X$. Choose $\epsilon =1$. Then, there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<1.$$

Hence, for all $\alpha \ge {\alpha }_0$ and every $x\in B$, we have

$$\left|f_{\alpha }\left(x\right)\right| \le \left|f\left(x\right)\right| + \left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\left|f\left(x\right)\right| + 1.$$

Taking the supremum over $x\in B$, we obtain

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right|<\underset{x\in B}{sup}\left|f\left(x\right)\right| + 1\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_0.$$

For the finitely many indices $\alpha$ with $\alpha <{\alpha }_0$, each functional $f_{\alpha }$ is $\lambda$-continuous and thus bounded on B, that is,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right|<\infty \mathrm{for}\mathrm{all}\alpha <{\alpha }_0.$$

Define

$$M_0:=max\left\{\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right|:\alpha <{\alpha }_0\right\}.$$

Now, let

$$M_B:=max\left\{M_0,\underset{x\in B}{sup}\left|f\left(x\right)\right| + 1\right\}.$$

Then, for every index $\alpha$, we have

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right| \le M_B.$$

This establishes that the net $\left(f_{\alpha }\right)$ is uniformly bounded on the bounded set B.  □

Theorem 9 

(Characterization of Reflexivity). Let $(X,\lambda )$ be a locally solid convergence vector lattice such that every bounded subset of X is λ-complete. Then, $(X,\lambda )$ is reflexive if and only if for every λ-continuous linear functional $\varphi \in X^{**}$, there exists $x\in X$, satisfying

$$\varphi \left(f\right)=f\left(x\right)\mathit{for}\mathit{all}f\in X^{*}.$$

Proof. 

We prove the necessity and sufficiency with the following:

Necessity:

Suppose that $(X,\lambda )$ is reflexive. By definition, the canonical embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*},$$

is a $\lambda$-homeomorphism onto its image and $J\left(X\right)$ is dense in $X^{**}$ with respect to the dual convergence structure ${\lambda }^{*}$. Let $\varphi \in X^{**}$ be arbitrary. Since $J\left(X\right)$ is dense in $X^{**}$, there exists a net $\left(x_{\alpha }\right)$ in X such that

$$J\left(x_{\alpha }\right)\stackrel{{\lambda }^{*}}{\to }\varphi .$$

This means that for every bounded subset $B\subset X^{*}$,

$$\underset{f\in B}{sup}\left|J\left(x_{\alpha }\right)\left(f\right)-\varphi \left(f\right)\right|=\underset{f\in B}{sup}\left|f\left(x_{\alpha }\right)-\varphi \left(f\right)\right|\to 0.$$

In particular, taking B to be the unit ball of $X^{*}$ (which is bounded), we deduce that $\left(x_{\alpha }\right)$ is a Cauchy net in X with respect to $\lambda$. By the completeness of the bounded sets in X, there exists $x\in X$, such that

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

Since the canonical embedding J is $\lambda$-continuous, we have

$$J\left(x_{\alpha }\right)\stackrel{{\lambda }^{*}}{\to }J\left(x\right).$$

By the uniqueness of limits in the Hausdorff space $X^{**}$, it follows that

$$\varphi =J\left(x\right),$$

i.e., for every $f\in X^{*}$,

$$\varphi \left(f\right)=J\left(x\right)\left(f\right)=f\left(x\right).$$

Sufficiency:

Conversely, suppose that for every $\lambda$-continuous linear functional $\varphi \in X^{**}$, there exists $x\in X$ such that

$$\varphi \left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*}.$$

That is, every element of $X^{**}$ is of the form $J\left(x\right)$ for some $x\in X$, so that

$$X^{**}=J\left(X\right).$$

Since J is already known to be injective and $\lambda$-continuous, it now becomes a bijective $\lambda$-homeomorphism onto $X^{**}$. By definition, this means that $(X,\lambda )$ is reflexive.

We thus show that $(X,\lambda )$ is reflexive if and only if for every $\lambda$-continuous linear functional $\varphi \in X^{**}$, there exists $x\in X$, satisfying

$$\varphi \left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*}.$$

This completes the proof.  □

Proposition 8 

(Spectral Duality). Let $T:X\to X$ be a λ-continuous linear operator on a locally solid convergence vector lattice $(X,\lambda )$ and let $T^{*}:X^{*}\to X^{*}$ be its dual operator. Assume that T is power-bounded. Then, the spectral radii of T and $T^{*}$ coincide, i.e.,

$$r\left(T\right)=r\left(T^{*}\right).$$

Proof. 

Since T is $\lambda$-continuous, it follows that for every $n\ge 1$, the operator $T^n$ is also $\lambda$-continuous. The norm of $T^{*}$ satisfies the standard operator norm inequality:

$$\parallel T^{*}\parallel =\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*}\left(f\right)\parallel }{\parallel f\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{|f(Tx)|}{\parallel f\parallel \parallel x\parallel }}.$$

Applying this to the iterates $T^n$ and using the definition of $T^{*}$, we obtain

$$\parallel T^{*n}\parallel =\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*n}\left(f\right)\parallel }{\parallel f\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{|f(T^{n}x)|}{\parallel f\parallel \parallel x\parallel }} \le \underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{n}x\parallel }{\parallel x\parallel }}=\parallel T^n\parallel .$$

Taking the nth root on both sides and passing to the limit, we obtain

$$r\left(T^{*}\right)=\underset{n\to \infty }{lim}\parallel T^{*n}{\parallel }^{1/n} \le \underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n}=r\left(T\right).$$

To prove the reverse inequality, recall that if $\lambda$ is locally solid, then the dual pairing $\langle X,X^{*}\rangle$ respects the locally solid convergence structure. That is, for every $x\in X$ and $f\in X^{*}$,

$$|f\left(T^{n}x\right)| \le \parallel T^{*n}f\parallel \parallel x\parallel .$$

Taking the supremum over x in the unit ball of X, we obtain

$$\parallel T^n\parallel =\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{n}x\parallel }{\parallel x\parallel }} \le \underset{\begin{array}{c}x\ne 0\end{array}}{sup}\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{|f(T^{n}x)|}{\parallel f\parallel \parallel x\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*n}f\parallel }{\parallel f\parallel }}=\parallel T^{*n}\parallel .$$

Taking the nth root and passing to the limit, we obtain

$$r\left(T\right)=\underset{n\to \infty }{lim}\parallel T^n{\parallel }^{1/n} \le \underset{n\to \infty }{lim}{\parallel T^{*n}\parallel }^{1/n}=r\left(T^{*}\right).$$

Since we have established both inequalities,

$$r\left(T\right) \le r\left(T^{*}\right)\mathrm{and}r\left(T^{*}\right) \le r\left(T\right),$$

it follows that

$$r\left(T\right)=r\left(T^{*}\right).$$

This completes the proof.  □

Remark 9 

(Spectral Implications of Operator-Unbounded Convergence). Proposition 8 establishes that for a positive, power-bounded operator T between Banach lattices, the spectral radii satisfy

$$r\left(T\right)=r\left(T^{*}\right).$$

This equality reveals a deep connection between the unbounded convergence framework and spectral theory. In particular, the positivity of T ensures that the order structure is preserved, which is crucial in transferring convergence properties from T to its dual operator $T^{*}$. Consequently, the spectral radius, defined by

$$r\left(T\right)=\underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n},$$

remains invariant under taking duals.

This result not only highlights the robustness of the unbounded convergence approach in the study of Banach lattices but also provides additional insights into the behavior of positive operators. For instance, the spectral duality can be used to analyze the asymptotic behavior of iterates of T and, in turn, offers potential applications in ergodic theory and stability analysis. We note that similar spectral implications have been observed in classical results on positive operators (see, e.g., [17,18]). Further investigation along these lines could reveal additional connections between operator theory, spectral properties, and convergence structures in Banach lattices.

Theorem 10 

(Ergodic Duality). Let $(X,\lambda )$ be a reflexive locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous, power-bounded operator. We define the ergodic projection $P:X\to X$ by

$$P\left(x\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

Then, for every $f\in X^{*}$, the dual ergodic averages

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{\left(T^{*}\right)}^n\left(f\right)$$

converge in ${\lambda }^{*}$ to a functional $P^{*}\left(f\right)$, satisfying

$$f\left(P\left(x\right)\right)=P^{*}\left(f\right)\left(x\right)\mathit{for}\mathit{all}x\in X.$$

Proof. 

Let $T:X\to X$ be a $\lambda$-continuous positive linear operator on a locally solid vector lattice $(X,\lambda )$. Assume that the sequence of Cesàro averages

$$A_n\left(T\right)x:={\displaystyle \frac{1}{n}}\sum _{k=0}^{n-1}{T}^{k}x$$

converges in $\lambda$ for every $x\in X$. We aim to show that the sequence of Cesàro averages

$$A_n\left(T^{*}\right)f:={\displaystyle \frac{1}{n}}\sum _{k=0}^{n-1}{T}^{*k}f$$

also converges in the dual space $(X^{*},{\lambda }^{*})$ for every $f\in X^{*}$.

Since T is $\lambda$-continuous and positive, it follows that its dual operator $T^{*}:X^{*}\to X^{*}$ is also ${\lambda }^{*}$-continuous and positive. Given that $A_n\left(T\right)x$ converges in $\lambda$ for every $x\in X$, let

$$y=\underset{n\to \infty }{lim}{A}_n\left(T\right)x.$$

Since $\lambda$ is locally solid, the dual pairing $\langle X,X^{*}\rangle$ satisfies

$$f\left(A_n\left(T\right)x\right)\to f\left(y\right)\mathrm{for}\mathrm{every}f\in X^{*}.$$

Rewriting this in terms of $T^{*}$, we obtain

$$\langle A_n\left(T\right)x,f\rangle =\langle x,A_n\left(T^{*}\right)f\rangle .$$

Thus, for every $x\in X$ and $f\in X^{*}$,

$$\langle x,A_n\left(T^{*}\right)f\rangle \to \langle x,y\rangle .$$

(2)

Since x was arbitrary, this suggests that the sequence $\left(A_n\left(T^{*}\right)f\right)$ should converge in $X^{*}$.

To rigorously establish convergence in ${\lambda }^{*}$, we invoke the uniform boundedness principle in the dual space. The assumption that $A_n\left(T\right)x$ converges in $\lambda$ for every $x\in X$ implies that the sequence $\left\{A_n\left(T\right)\right\}$ is uniformly bounded in the operator norm, meaning that there exists a constant $C>0$ such that

$$\underset{n}{sup}\parallel A_n\left(T\right)\parallel \le C.$$

Since taking duals preserves uniform boundedness, it follows that

$$\underset{n}{sup}\parallel A_n\left(T^{*}\right)\parallel \le C.$$

Thus, the sequence $\left\{A_n\left(T^{*}\right)f\right\}$ is uniformly bounded for each $f\in X^{*}$.

By the Banach–Alaoglu theorem, every bounded sequence in $X^{*}$ has a ${\lambda }^{*}$-convergent subnet. Let g be a cluster point of $A_n\left(T^{*}\right)f$ in ${\lambda }^{*}$. This means that there exists a subnet $\left(A_{n_k}\left(T^{*}\right)f\right)$ such that

$$A_{n_k}\left(T^{*}\right)f\stackrel{{\lambda }^{*}}{\to }g.$$

From (2), we established that $\langle x,A_n\left(T^{*}\right)f\rangle$ converges for every $x\in X$. Since limits in the dual space are unique in the locally solid topology, we conclude that the entire net $A_n\left(T^{*}\right)f$ must converge to g. Hence,

$$A_n\left(T^{*}\right)f\stackrel{{\lambda }^{*}}{\to }g\mathrm{for}\mathrm{every}f\in X^{*}.$$

Since the Cesàro averages $A_n\left(T\right)x$ converge in $\lambda$ for every $x\in X$ and since the same reasoning applies to $A_n\left(T^{*}\right)$ in $X^{*}$ using the locally solid topology, we conclude that the ergodic theorem holds in the dual space:

$$A_n\left(T^{*}\right)f\stackrel{{\lambda }^{*}}{\to }g\mathrm{for}\mathrm{every}f\in X^{*}.$$

Thus, the Cesàro means of the dual operator $T^{*}$ also converge in ${\lambda }^{*}$.

This establishes the desired ergodic duality, completing the proof.  □

7. Non-Archimedean Dual Convergence Space

The results in this section establish a deep connection between the convergence properties of a locally solid convergence space and those of its dual. In particular, the equivalence of spectral radii and the ergodic duality theorem illustrate how reflexivity influences both spectral theory and the asymptotic behavior of operators. These findings open up new avenues for exploring nonlinear spectral theory and ergodic properties in ordered vector spaces.

Definition 19. 

Let $(X,\lambda )$ be a non-Archimedean locally solid convergence vector lattice over a non-Archimedean field K. Define the non-Archimedean dual space as

$$X_{NA}^{*}=\{f:X\to K\mid f\mathit{is}\mathit{linear}\mathit{and}\lambda \text{-}\mathit{continuous}\},$$

equipped with the topology of uniform convergence on ultrametrically bounded sets, that is, a net $\left(f_{\alpha }\right)$ in $X_{NA}^{*}$ converges to f if

$$\underset{x\in B}{sup}|f_{\alpha }\left(x\right)-f\left(x\right)|\to 0\mathit{for}\mathit{every}\mathit{ultrametrically}\mathit{bounded}\mathit{set}B\subset X.$$

Definition 20 

(Choquet-Modified Dual Convergence Structure). Let $(X,\lambda )$ be a locally solid convergence vector lattice and let μ be a Choquet capacity defined on X. The Choquet-modified dual space is defined as

$$X_{Ch}^{*}=\{f:X\to \mathbb{R}\left(\mathit{or}\mathbb{C}\right)\mid f\mathit{is}\mathit{linear}\mathit{and}\lambda \text{-}\mathit{continuous}\},$$

with a convergence structure where a net $\left(f_{\alpha }\right)$ converges to f if for every bounded set $B\subset X$,

$$\mu \left(|f_{\alpha }-f|\right)\to 0.$$

Lemma 9 

(Uniform Boundedness in the Non-Archimedean Dual). Let $(X,\lambda )$ be a non-Archimedean locally solid convergence space and let $\left(f_{\alpha }\right)$ be a net in $X^{*}$ converging in the above sense. Then, for every ultrametrically bounded set $B\subset X$, there exists a constant $M_B>0$ such that

$$\underset{\alpha }{sup}\underset{x\in B}{sup}|f_{\alpha }\left(x\right)| \le M_B.$$

Proof. 

Let $\left(f_{\alpha }\right)$ be a net in the dual space $X^{*}$ of a non-Archimedean locally solid vector lattice $(X,\lambda )$. Suppose that $\left(f_{\alpha }\right)$ converges to some $f\in X^{*}$ with respect to the dual convergence structure ${\lambda }^{*}$, meaning that for every bounded subset $B\subset X$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|\to 0\mathrm{as}\alpha \to \infty .$$

We wish to show that the net $\left(f_{\alpha }\right)$ is uniformly bounded on bounded sets.

Fix a bounded subset $B\subset X$. Since $\left(f_{\alpha }\right)$ converges to f in ${\lambda }^{*}$, for every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha \ge {\alpha }_0$,

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\epsilon .$$

Taking $\epsilon =1$, it follows that for all $\alpha \ge {\alpha }_0$ and every $x\in B$,

$$\left|f_{\alpha }\left(x\right)\right| \le \left|f\left(x\right)\right| + \left|f_{\alpha }\left(x\right)-f\left(x\right)\right|<\left|f\left(x\right)\right| + 1.$$

Taking the supremum over $x\in B$, we obtain

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right|<\underset{x\in B}{sup}\left|f\left(x\right)\right| + 1\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_0.$$

For the finitely many indices $\alpha <{\alpha }_0$, each functional $f_{\alpha }$ is $\lambda$-continuous and thus bounded on B, meaning that there exists some finite constant $M_0$ such that

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right| \le M_0\mathrm{for}\mathrm{all}\alpha <{\alpha }_0.$$

We define

$$M_B:=max\left\{M_0,\underset{x\in B}{sup}\left|f\left(x\right)\right| + 1\right\}.$$

Then, for all $\alpha$, we have

$$\underset{x\in B}{sup}\left|f_{\alpha }\left(x\right)\right| \le M_B.$$

Thus, the net $\left(f_{\alpha }\right)$ is uniformly bounded on the bounded set B.

Since B was an arbitrary bounded set in X, we conclude that $\left(f_{\alpha }\right)$ is uniformly bounded on all bounded subsets of X. This proves the uniform boundedness principle in the non-Archimedean dual.  □

Theorem 11 

(Non-Archimedean Spectral Duality). Let $T:X\to X$ be a λ-continuous linear operator on a non-Archimedean locally solid convergence space $(X,\lambda )$ and let $T^{*}:X^{*}\to X^{*}$ be its dual. If T is power-bounded, then the spectral radii satisfy

$$r\left(T\right)=r\left(T^{*}\right).$$

Proof. 

Let $T:X\to X$ be a $\lambda$-continuous positive linear operator on a non-Archimedean locally solid vector lattice $(X,\lambda )$ and let $T^{*}:X^{*}\to X^{*}$ be its dual operator, defined by

$$T^{*}\left(f\right)\left(x\right)=f\left(Tx\right)\mathrm{for}\mathrm{all}f\in X^{*},x\in X.$$

We aim to show that the spectral radii of T and $T^{*}$ coincide, i.e.,

$$r\left(T\right)=r\left(T^{*}\right),$$

where the spectral radius of an operator S is defined as

$$r\left(S\right)=\underset{n\to \infty }{lim}{\parallel S^n\parallel }^{1/n}.$$

Since T is $\lambda$-continuous, it follows that for every $n\ge 1$, the iterates $T^n$ are also $\lambda$-continuous. The norm of $T^{*}$ satisfies the standard operator norm inequality:

$$\parallel T^{*}\parallel =\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*}\left(f\right)\parallel }{\parallel f\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{|f(Tx)|}{\parallel f\parallel \parallel x\parallel }}.$$

Applying this to the iterates $T^n$ and using the definition of $T^{*}$, we obtain

$$\parallel T^{*n}\parallel =\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*n}\left(f\right)\parallel }{\parallel f\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{|f(T^{n}x)|}{\parallel f\parallel \parallel x\parallel }} \le \underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{n}x\parallel }{\parallel x\parallel }}=\parallel T^n\parallel .$$

Taking the nth root on both sides and passing to the limit, we obtain

$$r\left(T^{*}\right)=\underset{n\to \infty }{lim}\parallel T^{*n}{\parallel }^{1/n} \le \underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n}=r\left(T\right).$$

To prove the reverse inequality, we recall that in a non-Archimedean locally solid vector lattice, the dual pairing $\langle X,X^{*}\rangle$ respects the non-Archimedean locally solid convergence structure. That is, for every $x\in X$ and $f\in X^{*}$,

$$|f\left(T^{n}x\right)| \le \parallel T^{*n}f\parallel \parallel x\parallel .$$

Taking the supremum over x in the unit ball of X, we obtain

$$\parallel T^n\parallel =\underset{\begin{array}{c}x\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{n}x\parallel }{\parallel x\parallel }} \le \underset{\begin{array}{c}x\ne 0\end{array}}{sup}\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{|f(T^{n}x)|}{\parallel f\parallel \parallel x\parallel }}=\underset{\begin{array}{c}f\ne 0\end{array}}{sup}{\displaystyle \frac{\parallel T^{*n}f\parallel }{\parallel f\parallel }}=\parallel T^{*n}\parallel .$$

Taking the nth root and passing to the limit, we obtain

$$r\left(T\right)=\underset{n\to \infty }{lim}\parallel T^n{\parallel }^{1/n} \le \underset{n\to \infty }{lim}{\parallel T^{*n}\parallel }^{1/n}=r\left(T^{*}\right).$$

Since we have established both inequalities,

$$r\left(T\right) \le r\left(T^{*}\right)\mathrm{and}r\left(T^{*}\right) \le r\left(T\right),$$

it follows that

$$r\left(T\right)=r\left(T^{*}\right).$$

Thus, the spectral radius remains invariant under duality in the non-Archimedean setting, completing the proof.  □

The extension of the duality framework to non-Archimedean and Choquet-modified settings introduces new phenomena due to the ultrametric and capacity-induced constraints. These refinements promise sharper convergence and stabilization properties that may yield novel insights into spectral and ergodic behavior.

Definition 21 

(p-adic Reflexive Convergence Space). Let $(X,{\lambda }_p)$ be a locally solid convergence vector lattice over the field of p-adic numbers ${\mathbb{Q}}_p$, equipped with the p-adic norm ${|·|}_p$. The space $(X,{\lambda }_p)$ is called p-adic reflexive if the canonical embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)(f\in X^{*})$$

is surjective with respect to the dual convergence structure induced by ${\lambda }_p$.

Definition 22 

(p-adic Nonlinear Operator). A mapping $T:X\to X$ on a p-adic locally solid convergence space $(X,{\lambda }_p)$ is called a p-adic nonlinear operator if it is ${\lambda }_p$-continuous and satisfies a local contraction condition on ultrametrically bounded subsets of X.

Lemma 10 

(p–adic Stability under Nonlinear Contraction). Let $T:X\to X$ be a ${\lambda }_p$-continuous nonlinear operator satisfying for some $0 \le k<1$ and every ultrametrically bounded regulator $u\in X^{+}$

$${|T\left(x\right)-T\left(y\right)|}_p \wedge u \le k\left({|x-y|}_p \wedge u\right) \mathit{for} \mathit{all} x,y\in X.$$

Then, the Picard iterates starting from any $x_0\in X$ converge uniformly on ultrametrically bounded subsets.

Proof. 

Let $(X,d)$ be a complete non-Archimedean metric space and let $T:X\to X$ be a nonlinear contraction, meaning that there exists a constant $c\in (0,1)$ such that

$$d(Tx,Ty) \le c·d(x,y) \mathrm{for} \mathrm{all} x,y\in X.$$

We aim to show that T has a unique fixed point and that iterations of T exhibit p-adic stability.

Consider an arbitrary point $x_0\in X$ and define the iterative sequence

$$x_{n+1}=T\left(x_n\right),\hspace{1em}n\ge 0.$$

We can show that $\left(x_n\right)$ is a Cauchy sequence. For any $m>n$, applying the contraction property iteratively gives

$$d(x_n,x_{n + 1})=d(T{x}_{n-1},T{x}_n) \le c·d(x_{n-1},x_n).$$

Expanding this recurrence, we obtain

$$d(x_n,x_{n + 1}) \le c^{n}d(x_0,x_1).$$

Summing over all $k\ge n$,

$$d(x_n,x_m) \le \sum _{k=n}^{m-1}d(x_k,x_{k + 1}) \le \sum _{k=n}^{m-1}{c}^{k}d(x_0,x_1).$$

Since $0<c<1$, the geometric series converges, implying that $\left(x_n\right)$ is Cauchy. Since X is complete, there exists a limit $x^{*}\in X$ such that

$$\underset{n\to \infty }{lim}{x}_n=x^{*}.$$

Applying T to both sides and using continuity, we obtain

$$T\left(x^{*}\right)=\underset{n\to \infty }{lim}T\left(x_n\right)=\underset{n\to \infty }{lim}{x}_{n+1}=x^{*}.$$

Thus, $x^{*}$ is a fixed point of T.

To prove uniqueness, assume $y^{*}$ is another fixed point. Then,

$$d(x^{*},y^{*})=d(T{x}^{*},T{y}^{*}) \le c·d(x^{*},y^{*}).$$

Since $0<c<1$, the only possibility is $d(x^{*},y^{*})=0$, implying $x^{*}=y^{*}$. Hence, $x^{*}$ is the unique fixed point.

To establish stability, let ${\tilde{x}}_0\in X$ be a perturbed initial value with corresponding iterates

$${\tilde{x}}_{n+1}=T\left({\tilde{x}}_n\right).$$

We estimate the deviation:

$$d(x_n,{\tilde{x}}_n)=d(T{x}_{n-1},T{\tilde{x}}_{n-1}) \le c·d(x_{n-1},{\tilde{x}}_{n-1}).$$

Iterating this bound, we obtain

$$d(x_n,{\tilde{x}}_n) \le c^{n}d(x_0,{\tilde{x}}_0).$$

Since $c^n\to 0$ as $n\to \infty$, it follows that

$$\underset{n\to \infty }{lim}d(x_n,{\tilde{x}}_n)=0.$$

Thus, the iterations of T are stable under small perturbations in the non-Archimedean metric.

We have shown that T has a unique fixed point $x^{*}$ and the iteration sequence $\left(x_n\right)$ converges to $x^{*}$ with stability under perturbations in the non-Archimedean metric. This completes the proof.  □

Theorem 12 

(p-adic Nonlinear Reflexivity). Let $(X,{\lambda }_p)$ be a complete p-adic locally solid convergence space and let $T:X\to X$ be a ${\lambda }_p$-continuous nonlinear operator satisfying the contraction condition of the previous lemma. Then, the canonical embedding

$$J:X\to X^{**}$$

is surjective, i.e., X is reflexive in the p-adic sense.

Proof. 

Let $(X,d)$ be a complete non-Archimedean metric space and let $T:X\to X$ be a nonlinear mapping satisfying a reflexivity condition. Specifically, assume that for every bounded sequence $\left(x_n\right)\subset X$, the iterates $\left(T^n{x}_n\right)$ have a cluster point in X.

We aim to show that T has a fixed point in X.

Since X is complete and T satisfies the reflexivity condition, let $\left(x_n\right)$ be an arbitrary bounded sequence in X. By assumption, the iterates $\left(T^n{x}_n\right)$ have a cluster point, say $x^{*}\in X$, meaning that there exists a subsequence $\left(T^{n_k}{x}_{n_k}\right)$ such that

$$\underset{k\to \infty }{lim}{T}^{n_k}{x}_{n_k}=x^{*}.$$

Since T is a continuous mapping (as assumed in the reflexivity setting), we apply T to both sides of the convergence equation:

$$T\left(\underset{k\to \infty }{lim}{T}^{n_k}{x}_{n_k}\right)=\underset{k\to \infty }{lim}{T}^{n_k + 1}{x}_{n_k}.$$

Since $T^{n_k}{x}_{n_k}\to x^{*}$, it follows that

$$T\left(x^{*}\right)=x^{*}.$$

Thus, $x^{*}$ is a fixed point of T.

To show uniqueness, suppose that there exists another fixed point $y^{*}$ such that

$$T\left(y^{*}\right)=y^{*}.$$

Using the non-Archimedean property of the metric d, we estimate

$$d(T^n{x}^{*},T^n{y}^{*})=d(x^{*},y^{*}).$$

Since $x^{*}$ and $y^{*}$ are both fixed points of T, we obtain

$$d(x^{*},y^{*})=d(T\left(x^{*}\right),T\left(y^{*}\right))=d(x^{*},y^{*}).$$

In a non-Archimedean metric, this implies that $x^{*}=y^{*}$, proving uniqueness.

We have shown that T has a unique fixed point in X under the given reflexivity condition. This establishes the p-adic nonlinear reflexivity of T.  □

The p-adic setting introduces additional structure that facilitates the study of nonlinear operators. The interplay between ultrametric contraction and reflexivity enriches the theory of nonlinear dynamics in ordered vector spaces, suggesting new applications in p-adic differential equations and ergodic theory.

Definition 23 

(Uniformly Contractive Dual Operator). Let $T:X\to X$ be a λ-continuous operator on a reflexive locally solid convergence space $(X,\lambda )$ that is uniformly contractive, i.e., there exists $0 \le k<1$ such that

$$|T\left(x\right)-T\left(y\right)| \wedge u \le k(|x-y| \wedge u)\mathit{for}\mathit{all}x,y\in X\mathit{and}\mathit{every}\mathit{regulator}u\in X^{+}.$$

Then, the dual operator

$$T^{*}:X^{*}\to X^{*}$$

is said to be uniformly contractive if it satisfies an analogous inequality with respect to the dual convergence.

Proposition 9 

(Fixed Point Duality). Let $T:X\to X$ be a uniformly contractive λ-continuous operator on a reflexive locally solid convergence space $(X,\lambda )$. Then,

1. 

T has a unique fixed point $x^{*}\in X$.

2. 

The dual operator $T^{*}:X^{*}\to X^{*}$ is uniformly contractive.

3. 

For every $f\in X^{*}$, the fixed point satisfies

$$f\left(x^{*}\right)=T^{*}\left(f\right)\left(x^{*}\right).$$

Proof. 

We will prove each part separately.

(1) Since T is uniformly contractive, there exists a constant $c\in (0,1)$ such that

$$d(Tx,Ty) \le c·d(x,y)\mathrm{for}\mathrm{all}x,y\in X.$$

Consider an arbitrary starting point $x_0\in X$ and define the iterative sequence:

$$x_{n+1}=T\left(x_n\right),n\ge 0.$$

Applying the contractive property repeatedly gives

$$d(x_n,x_{n + 1})=d(T{x}_{n-1},T{x}_n) \le c·d(x_{n-1},x_n).$$

By induction, we obtain

$$d(x_n,x_{n + 1}) \le c^{n}d(x_0,x_1).$$

Summing over all $k\ge n$,

$$d(x_n,x_m) \le \sum _{k=n}^{m-1}{c}^{k}d(x_0,x_1).$$

Since $c^k$ forms a geometric series and $0<c<1$, the series converges, implying that $\left(x_n\right)$ is Cauchy. By completeness of X, there exists a limit $x^{*}\in X$ such that

$$\underset{n\to \infty }{lim}{x}_n=x^{*}.$$

Applying T to both sides and using continuity, we obtain

$$T\left(x^{*}\right)=\underset{n\to \infty }{lim}T\left(x_n\right)=\underset{n\to \infty }{lim}{x}_{n+1}=x^{*}.$$

Thus, $x^{*}$ is a fixed point of T.

To prove uniqueness, assume that there is another fixed point $y^{*}$. Then,

$$d(x^{*},y^{*})=d(T{x}^{*},T{y}^{*}) \le c·d(x^{*},y^{*}).$$

Since $0<c<1$, the only possibility is $d(x^{*},y^{*})=0$, implying that $x^{*}=y^{*}$. Thus, T has a unique fixed point.

(2) The dual operator $T^{*}:X^{*}\to X^{*}$ is defined by

$$T^{*}\left(f\right)\left(x\right)=f\left(Tx\right)\mathrm{for}\mathrm{all}f\in X^{*},x\in X.$$

Since T is uniformly contractive, for any $f,g\in X^{*}$ and all $x\in X$, we have

$$|T^{*}\left(f\right)\left(x\right)-T^{*}\left(g\right)\left(x\right)| = |f\left(Tx\right)-g\left(Tx\right)|.$$

Taking the supremum over the unit ball in X, we obtain

$$\parallel T^{*}\left(f\right)-T^{*}\left(g\right)\parallel =\underset{\parallel x\parallel \le 1}{sup}|f\left(Tx\right)-g\left(Tx\right)|.$$

Using the contraction property of T,

$$\parallel T^{*}\left(f\right)-T^{*}\left(g\right)\parallel \le c\underset{\parallel x\parallel \le 1}{sup}|f\left(x\right)-g\left(x\right)|=c\parallel f-g\parallel .$$

Thus, $T^{*}$ is also uniformly contractive with the same constant $c\in (0,1)$.

(3) For every $f\in X^{*}$, we have

$$T^{*}\left(f\right)\left(x^{*}\right)=f\left(T\left(x^{*}\right)\right)=f\left(x^{*}\right).$$

Thus, the fixed point $x^{*}$ satisfies

$$f\left(x^{*}\right)=T^{*}\left(f\right)\left(x^{*}\right).$$

This completes the proof.  □

Lemma 11 

(Dual Convergence of Picard Iterates). Under the hypotheses of the preceding proposition, the Picard iteration sequence $\left\{x_n\right\}$ defined by $x_{n + 1}=T\left(x_n\right)$ converges in λ and, for every $f\in X^{*}$, the sequence $\left\{f\left(x_n\right)\right\}$ converges uniformly.

Proof. 

Since $T:X\to X$ is a uniformly contractive $\lambda$-continuous operator, there exists a constant $c\in (0,1)$ such that

$$d(Tx,Ty) \le c·d(x,y)\mathrm{for}\mathrm{all}x,y\in X.$$

Consider the Picard iteration sequence defined by

$$x_{n+1}=T\left(x_n\right),n\ge 0.$$

Applying the contraction property iteratively gives

$$d(x_n,x_{n + 1})=d(T{x}_{n-1},T{x}_n) \le c·d(x_{n-1},x_n).$$

By induction, we obtain

$$d(x_n,x_{n + 1}) \le c^{n}d(x_0,x_1).$$

Summing over all $k\ge n$,

$$d(x_n,x_m) \le \sum _{k=n}^{m-1}{c}^{k}d(x_0,x_1).$$

Since $0<c<1$, the geometric series converges, implying that $\left(x_n\right)$ is a Cauchy sequence. By the completeness of X, there exists a limit $x^{*}\in X$ such that

$$\underset{n\to \infty }{lim}{x}_n=x^{*}.$$

Since T is $\lambda$-continuous and $\left(x_n\right)$ is a Cauchy sequence, for every regulator $u\in X^{+}$, we have

$$|x_{n+1}-x_n| \wedge u\stackrel{\lambda }{\to }0.$$

Since $\lambda$ is locally solid, the lattice structure ensures that the sequence $\left(x_n\right)$ converges to $x^{*}$ in $\lambda$.

For every $f\in X^{*}$, applying f to the iteration sequence, we obtain

$$|f\left(x_{n + 1}\right)-f\left(x_n\right)| = |f\left(T{x}_n\right)-f\left(T{x}_{n-1}\right)|.$$

Using the contraction property,

$$|f\left(x_{n + 1}\right)-f\left(x_n\right)| \le c|f\left(x_n\right)-f\left(x_{n-1}\right)|.$$

By induction, we obtain

$$|f\left(x_n\right)-f\left(x_{n-1}\right)| \le c^n|f\left(x_1\right)-f\left(x_0\right)|.$$

Summing over all $k\ge n$,

$$|f\left(x_n\right)-f\left(x_m\right)| \le \sum _{k=n}^{m-1}{c}^k|f\left(x_1\right)-f\left(x_0\right)|.$$

Since $c^k$ forms a geometric series, the series converges, implying that $\left(f\left(x_n\right)\right)$ is a Cauchy sequence. By the completeness of $\mathbb{R}$ (or $\mathbb{C}$), there exists a limit:

$$\underset{n\to \infty }{lim}f\left(x_n\right)=f\left(x^{*}\right).$$

Thus, the sequence $\left\{f\left(x_n\right)\right\}$ converges uniformly on bounded sets.

This completes the proof.  □

Theorem 13 

(Fixed Point Duality Theorem). Let $T:X\to X$ be as above. Then, the unique fixed point $x^{*}$ of T satisfies

$$f\left(x^{*}\right)=\underset{n\to \infty }{lim}f\left(x_n\right)=\underset{n\to \infty }{lim}{T}^{*}\left(f\right)\left(x_n\right)=T^{*}\left(f\right)\left(x^{*}\right)$$

for every $f\in X^{*}$.

Proof. 

Let $\left\{x_n\right\}$ be the Picard iteration sequence defined by

$$x_{n+1}=T\left(x_n\right),n\ge 0.$$

By the contraction property of T, we know from the previous lemma that $\left(x_n\right)$ converges in $\lambda$ to the unique fixed point $x^{*}$, i.e.,

$$\underset{n\to \infty }{lim}{x}_n=x^{*}.$$

For any $f\in X^{*}$, applying f to the sequence $\left(x_n\right)$ and using the uniform convergence result from the previous lemma, we obtain

$$\underset{n\to \infty }{lim}f\left(x_n\right)=f\left(x^{*}\right).$$

Thus, the functional values $f\left(x_n\right)$ converge to $f\left(x^{*}\right)$.

Since $T^{*}$ is the dual operator, we apply it to f:

$$T^{*}\left(f\right)\left(x_n\right)=f\left(T{x}_n\right)=f\left(x_{n + 1}\right).$$

Taking limits on both sides and using the fact that $x_n\to x^{*}$, we obtain

$$\underset{n\to \infty }{lim}{T}^{*}\left(f\right)\left(x_n\right)=\underset{n\to \infty }{lim}f\left(x_{n + 1}\right)=f\left(x^{*}\right).$$

Since $x^{*}$ is the unique fixed point of T, we have

$$T\left(x^{*}\right)=x^{*}.$$

Applying f to both sides gives

$$f\left(T\left(x^{*}\right)\right)=f\left(x^{*}\right).$$

By the definition of $T^{*}$, this can be rewritten as

$$T^{*}\left(f\right)\left(x^{*}\right)=f\left(x^{*}\right).$$

We thus establish the desired sequence of equalities:

$$f\left(x^{*}\right)=\underset{n\to \infty }{lim}f\left(x_n\right)=\underset{n\to \infty }{lim}{T}^{*}\left(f\right)\left(x_n\right)=T^{*}\left(f\right)\left(x^{*}\right).$$

Thus, the dual operator preserves the fixed point under evaluation, completing the proof.  □

This interplay between duality and fixed point theory offers a robust framework for analyzing iterative methods in nonlinear settings. The uniform contraction condition serves as a bridge between the behavior of operators in the primal and dual spaces, with significant implications for convergence analysis in applications ranging from differential equations to ergodic theory.

In many applications, one studies vector lattices over non-Archimedean fields (i.e., fields equipped with an ultrametric absolute value satisfying the strong triangle inequality). In such settings, the classical Archimedean property is not assumed. This necessitates a modification of the theory of locally solid convergence structures.

Definition 24 

(Non-Archimedean Vector Lattice). A vector lattice X over a non-Archimedean field K (i.e., a field equipped with a non-Archimedean absolute value $|·|$ satisfying

$$|x + y| \le max\{|x|,|y|\}\mathit{for}\mathit{all}x,y\in K)$$

is said to be non-Archimedean if it is not required that for every $x,y\in X_{+}$ the condition

$$nx \le y\mathit{for}\mathit{all}n\in \mathbb{N}$$

implies $x=0$. That is, the usual Archimedean property is relaxed.

Definition 25 

(Non-Archimedean Locally Solid Convergence). Let X be a non-Archimedean vector lattice and let λ be a convergence structure on X. We say that λ is non-Archimedean locally solid if

1. 

For every net $\left(x_{\alpha }\right)$ in X with $x_{\alpha }\stackrel{\lambda }{\to }0$ and every net $\left(y_{\alpha }\right)$ satisfying $|y_{\alpha }| \le |x_{\alpha }|$ for all α, we have $y_{\alpha }\stackrel{\lambda }{\to }0$.

2. 

The ultrametric inequality inherent in the non-Archimedean norm is compatible with the convergence structure, that is, for every $\epsilon >0$, there exists a λ-neighborhood U of zero such that if $x,y\in U$, then $|x-y|<\epsilon$.

Lemma 12 

(Cauchy Nets in the Non-Archimedean Setting). Let X be a non-Archimedean locally solid vector lattice. Then, every λ-Cauchy net in X is eventually constant on any bounded subset of X.

Proof. 

Let $\left(x_{\alpha }\right)$ be a $\lambda$-Cauchy net in X. By definition, for every $\lambda$-neighborhood U of zero, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$x_{\alpha }-x_{\beta }\in U.$$

Since X is a non-Archimedean space, the ultrametric inequality holds, meaning that for any three elements $x,y,z\in X$,

$$d(x,z) \le max\left\{d\right(x,y),d(y,z\left)\right\}.$$

Applying this to the net, for all $\alpha ,\beta ,\gamma \ge {\alpha }_0$, we obtain

$$d(x_{\alpha },x_{\gamma }) \le max\{d(x_{\alpha },x_{\beta }),d(x_{\beta },x_{\gamma })\}.$$

Since $\left(x_{\alpha }\right)$ is a Cauchy net, the right-hand side becomes arbitrarily small, implying that the net eventually stabilizes.

Let B be a bounded subset of X. By the definition of boundedness in a non-Archimedean locally solid vector lattice, there exists a regulator $u\in X^{+}$ such that

$$|x_{\alpha }| \le u\mathrm{for}\mathrm{all}\mathrm{sufficiently}\mathrm{large}\alpha .$$

Because $\left(x_{\alpha }\right)$ is $\lambda$-Cauchy, for sufficiently large $\alpha ,\beta$, we have

$$|x_{\alpha }-x_{\beta }| \le u.$$

Since the non-Archimedean absolute value satisfies the strong triangle inequality, this implies that once the net enters a sufficiently small neighborhood, it remains constant on B.

Thus, every $\lambda$-Cauchy net $\left(x_{\alpha }\right)$ in X is eventually constant on any bounded subset of X, proving the claim.  □

Theorem 14 

(Monotone Convergence in Non-Archimedean Vector Lattices). Let X be a non-Archimedean locally solid vector lattice and let $\left(x_{\alpha }\right)$ be an increasing (monotone) net in X that is λ-Cauchy and eventually bounded. Then, $\left(x_{\alpha }\right)$ converges in the non-Archimedean locally solid convergence structure to its supremum.

Proof. 

Since $\left(x_{\alpha }\right)$ is an increasing net and eventually bounded, it has a supremum

$$x=\underset{\alpha }{sup}{x}_{\alpha }.$$

We must show that $x_{\alpha }\to x$ in the $\lambda$-topology.

Since $\left(x_{\alpha }\right)$ is $\lambda$-Cauchy, for every $\lambda$-neighborhood U of zero, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$x_{\alpha }-x_{\beta }\in U.$$

Using the ultrametric inequality, for any three indices $\alpha ,\beta ,\gamma$,

$$d(x_{\alpha },x_{\gamma }) \le max\{d(x_{\alpha },x_{\beta }),d(x_{\beta },x_{\gamma })\}.$$

Since the net is Cauchy, the right-hand side becomes arbitrarily small, implying that the differences $x_{\alpha }-x_{\beta }$ eventually vanish in the $\lambda$-topology.

For any regulator $u\in X^{+}$, since $x_{\alpha }\uparrow x$, the difference $x-x_{\alpha }$ is decreasing. By the non-Archimedean locally solid structure, we know that the differences

$$(x-x_{\alpha }) \wedge u$$

must eventually become arbitrarily small. Since $\lambda$ is locally solid, this implies that

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

Thus, the increasing $\lambda$-Cauchy net $\left(x_{\alpha }\right)$ converges to its supremum x in the non-Archimedean locally solid convergence structure.  □

Definition 26 

(Non-Archimedean Unbounded Convergence). Let X be a non-Archimedean locally solid vector lattice with convergence structure λ. The non-Archimedean unbounded modification $u\lambda$ is defined by

$$x_{\alpha }\stackrel{u\lambda }{\to }x\mathit{if}\mathit{and}\mathit{only}\mathit{if}|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0\mathit{for}\mathit{all}u\in X_{+}.$$

Here, the infimum is taken in the lattice sense and interpreted with respect to the ultrametric norm.

Proposition 10. 

Let X be a non-Archimedean locally solid vector lattice and let λ be a non-Archimedean locally solid convergence structure on X. Then, its unbounded modification $u\lambda$ is also non-Archimedean locally solid.

Proof. 

We must verify that the defining properties of a non-Archimedean locally solid convergence structure are preserved when passing from $\lambda$ to its unbounded modification $u\lambda$.

By definition, a net $\left(x_{\alpha }\right)$ in X is said to converge to x in $u\lambda$ if for every regulator $u\in X^{+}$,

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0.$$

We need to show that $u\lambda$ satisfies the following: 1. Local Solidity: If $|y_{\alpha }| \le |x_{\alpha }|$ and $x_{\alpha }\stackrel{u\lambda }{\to }x$, then $y_{\alpha }\stackrel{u\lambda }{\to }y$. 2. Non-Archimedean Property: The ultrametric inequality holds under $u\lambda$.

Assume that $x_{\alpha }\stackrel{u\lambda }{\to }x$, meaning that for every $u\in X^{+}$,

$$|x_{\alpha }-x| \wedge u\stackrel{\lambda }{\to }0.$$

Now, suppose that $|y_{\alpha }| \le |x_{\alpha }|$ for all $\alpha$. Since $\lambda$ is locally solid, this implies that

$$|y_{\alpha }-y| \wedge u \le |x_{\alpha }-x| \wedge u.$$

By the local solidity of λ, if the right-hand side converges to zero, then so does the left-hand side. Hence,

$$|y_{\alpha }-y| \wedge u\stackrel{\lambda }{\to }0.$$

Thus, $y_{\alpha }\stackrel{u\lambda }{\to }y$, proving that $u\lambda$ is locally solid.

In a non-Archimedean space, the ultrametric inequality states that for all $x,y,z\in X$,

$$d(x,z) \le max\left\{d\right(x,y),d(y,z\left)\right\}.$$

Applying this property to $u\lambda$, for any regulator $u\in X^{+}$,

$$|x_{\alpha }-x| \wedge u \le max\{|x_{\alpha }-x_{\beta }| \wedge u,|x_{\beta }-x| \wedge u\}.$$

Since both terms on the right-hand side tend to zero under $\lambda$, the left-hand side must also tend to zero. Thus, $u\lambda$ retains the non-Archimedean ultrametric property.

Since $u\lambda$ satisfies both local solidity and the non-Archimedean ultrametric inequality, it is a non-Archimedean locally solid convergence structure. Thus, the unbounded modification of $\lambda$ preserves its fundamental properties.  □

The extension to non-Archimedean settings uncovers new phenomena not observed in the classical Archimedean case. In particular, the strong ultrametric inequality forces every Cauchy net to become eventually constant on bounded sets, which is a striking difference compared to the Archimedean framework.

These preliminary results suggest several promising avenues for further research. For example, one may investigate duality and spectral theory in non-Archimedean Banach lattices, as well as the interplay between non-Archimedean locally solid convergence and p-adic analysis. For further background on non-Archimedean analysis, see [13].

8. Non-Archimedean Analysis

In this section, we extend the theory of locally solid convergence structures to the non-Archimedean setting. In particular, we focus on ultrametric spaces, p-adic analysis, duality, and the behavior of Cauchy nets. The non-Archimedean framework offers unique analytical properties that differ significantly from the classical Archimedean case.

Definition 27 

(Ultrametric Locally Solid Convergence). Let K be a non-Archimedean field endowed with an ultrametric absolute value $|·|$. A vector lattice X over K is said to admit an ultrametric locally solid convergence structure λ if

1. 

For every net $\left(x_{\alpha }\right)$ in X with $x_{\alpha }\stackrel{\lambda }{\to }0$ and every net $\left(y_{\alpha }\right)$ satisfying $|y_{\alpha }| \le |x_{\alpha }|$, it follows that $y_{\alpha }\stackrel{\lambda }{\to }0$.

2. 

There exists a basis of λ-neighborhoods of zero that are solid and satisfy the ultrametric inequality: for every $\epsilon >0$, there exists a neighborhood U such that

$$\forall x,y\in U,|x-y|<\epsilon .$$

This definition generalizes the classical notion of locally solid convergence to settings where the underlying field is non-Archimedean. The strong triangle (ultrametric) inequality profoundly influences the structure of convergent nets.

Definition 28 

(Dual Space in the Non-Archimedean Setting). Let $(X,\lambda )$ be a non-Archimedean locally solid convergence vector lattice over a non-Archimedean field K. The dual space $X^{*}$ is defined as the collection of all λ-continuous linear functionals

$$f:X\to K.$$

We equip $X^{*}$ with the topology of uniform convergence on ultrametrically bounded sets (i.e., subsets $B\subset X$, for which there exists a λ-neighborhood U of zero such that $B\subset U$).

Proposition 11 

(Completeness of the Non-Archimedean Dual). Let $(X,\lambda )$ be a complete non-Archimedean locally solid convergence vector lattice and assume that every ultrametrically bounded subset of X is λ-complete. Then, the dual space $X^{*}$, with the topology of uniform convergence on bounded sets, is complete.

Proof. 

Let $\left(f_{\alpha }\right)$ be a Cauchy net in $X^{*}$ with respect to the topology of uniform convergence on bounded sets. This means that for every bounded subset $B\subset X$ and every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$\underset{x\in B}{sup}|f_{\alpha }\left(x\right)-f_{\beta }\left(x\right)|<\epsilon .$$

We aim to show that there exists a functional $f\in X^{*}$ such that $f_{\alpha }\to f$ uniformly on bounded subsets of X.

Fix an arbitrary $x\in X$. Since $\left(f_{\alpha }\right)$ is Cauchy in $X^{*}$, the sequence $\left(f_{\alpha }\left(x\right)\right)$ is Cauchy in the non-Archimedean field K. By the completeness of K (with respect to its ultrametric norm), there exists a limit:

$$f\left(x\right):=\underset{\alpha }{lim}{f}_{\alpha }\left(x\right).$$

Thus, we define a mapping $f:X\to K$.

To show that f is linear, take any $x,y\in X$ and any scalar $a\in K$. Since each $f_{\alpha }$ is linear, we have

$$f_{\alpha }(x + y)=f_{\alpha }\left(x\right) + f_{\alpha }\left(y\right)\mathrm{and}{f}_{\alpha }\left(ax\right)=a{f}_{\alpha }\left(x\right).$$

Taking limits on both sides, using the continuity of limits in K, we obtain

$$f(x + y)=\underset{\alpha }{lim}{f}_{\alpha }(x + y)=\underset{\alpha }{lim}(f_{\alpha }\left(x\right) + f_{\alpha }\left(y\right))=f\left(x\right) + f\left(y\right),$$

$$f\left(ax\right)=\underset{\alpha }{lim}{f}_{\alpha }\left(ax\right)=\underset{\alpha }{lim}\left(a{f}_{\alpha }\left(x\right)\right)=af\left(x\right).$$

Thus, f is a linear functional on X.

Since $\left(f_{\alpha }\right)$ converges uniformly on every bounded set $B\subset X$, we have

$$\underset{x\in B}{sup}|f_{\alpha }\left(x\right)-f\left(x\right)|\to 0.$$

For any $\lambda$-Cauchy net $\left(x_{\beta }\right)$ in a bounded set, we have

$$\underset{\beta }{sup}|f_{\alpha }\left(x_{\beta }\right)-f_{\beta }\left(x_{\beta }\right)|\to 0.$$

Since $f_{\beta }\left(x_{\beta }\right)$ is Cauchy in K, the uniform limit $f\left(x_{\beta }\right)$ must also be Cauchy in K, implying that $f\left(x_{\beta }\right)\to f\left(x\right)$. This proves that f is $\lambda$-continuous.

For any bounded subset $B\subset X$ and any $\epsilon >0$, there exists ${\alpha }_0$, such that for all $\alpha \ge {\alpha }_0$,

$$\underset{x\in B}{sup}|f_{\alpha }\left(x\right)-f\left(x\right)|<\epsilon .$$

Thus, $f_{\alpha }\to f$ in the topology of uniform convergence on bounded sets.

Since every Cauchy net in $X^{*}$ has a limit in $X^{*}$, the space $X^{*}$, endowed with the topology of uniform convergence on bounded sets, is complete.  □

Lemma 13 

(Stabilization of Cauchy Nets). Let $(X,\lambda )$ be a non-Archimedean locally solid convergence vector lattice. Then, every λ-Cauchy net $\left(x_{\alpha }\right)$ in X stabilizes on every ultrametrically bounded subset. More precisely, for every bounded set $B\subset X$ and every $\epsilon >0$, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$ and all regulators $u\in B_{+}$,

$$|x_{\alpha }-x_{\beta }| \wedge u<\epsilon .$$

Proof. 

Since $\left(x_{\alpha }\right)$ is $\lambda$-Cauchy, for every $\lambda$-neighborhood U of zero, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$,

$$x_{\alpha }-x_{\beta }\in U.$$

We show that this implies uniform stabilization on bounded subsets.

Since X is a non-Archimedean space, it satisfies the ultrametric inequality, which states that for any three indices $\alpha ,\beta ,\gamma$,

$$d(x_{\alpha },x_{\gamma }) \le max\{d(x_{\alpha },x_{\beta }),d(x_{\beta },x_{\gamma })\}.$$

Applying this recursively to the Cauchy net $\left(x_{\alpha }\right)$, we conclude that once the net enters a sufficiently small $\lambda$-neighborhood, the differences $|x_{\alpha }-x_{\beta }|$ are uniformly small.

Let $B\subset X$ be an ultrametrically bounded subset. By definition, there exists a regulator $u\in X^{+}$ such that

$$|x_{\alpha }| \le u\mathrm{for}\mathrm{all}\mathrm{sufficiently}\mathrm{large}\alpha .$$

Since $\left(x_{\alpha }\right)$ is $\lambda$-Cauchy, for sufficiently large $\alpha ,\beta$, we have

$$|x_{\alpha }-x_{\beta }| \wedge u<\epsilon .$$

This ensures that the differences between elements of the net are uniformly small when observed through any regulator from $B_{+}$.

Since $|x_{\alpha }-x_{\beta }| \wedge u$ remains arbitrarily small for all regulators $u\in B_{+}$ and all sufficiently large $\alpha ,\beta$, the net stabilizes on every bounded subset. This means that for any bounded subset B, there exists an index ${\alpha }_0$ such that for all $\alpha ,\beta \ge {\alpha }_0$ and all $u\in B_{+}$,

$$|x_{\alpha }-x_{\beta }| \wedge u<\epsilon .$$

Thus, the Cauchy net $\left(x_{\alpha }\right)$ stabilizes on every ultrametrically bounded subset of X, completing the proof.  □

Definition 29 

(p-adic Locally Solid Convergence). Let p be a prime and consider the field of p-adic numbers ${\mathbb{Q}}_p$ endowed with the p-adic norm ${|·|}_p$. A vector lattice X over ${\mathbb{Q}}_p$ is said to have a p-adic locally solid convergence structure ${\lambda }_p$ if

1. 

For every net $\left(x_{\alpha }\right)$ in X converging to 0 in ${\lambda }_p$ and every net $\left(y_{\alpha }\right)$ satisfying $|y_{\alpha }{|}_p \le {|x_{\alpha }|}_p$, one has $y_{\alpha }\stackrel{{\lambda }_p}{\to }0$.

2. 

The convergence structure ${\lambda }_p$ is compatible with the p-adic ultrametric, i.e., for every $\epsilon >0$, there exists a ${\lambda }_p$-neighborhood U of zero such that if $x,y\in U$, then

$${|x-y|}_p<\epsilon .$$

The p-adic locally solid convergence structure provides a natural framework to study convergence phenomena in spaces over ${\mathbb{Q}}_p$. This setting is particularly significant in number theory and p-adic functional analysis, where the interplay between the order structure and the non-Archimedean norm reveals new duality properties.

Definition 30 

(Non-Archimedean Reflexivity). Let $(X,\lambda )$ be a non-Archimedean locally solid convergence vector lattice over a non-Archimedean field K. We say that X is reflexive if the natural embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)(f\in X^{*}),$$

is a λ-homeomorphism onto its image and if $J\left(X\right)$ is dense in $X^{**}$ with respect to the dual convergence structure ${\lambda }^{*}$.

Theorem 15 

(Characterization of Reflexivity). Assume that every ultrametrically bounded subset of X is λ-complete. Then, X is reflexive if and only if for every λ-continuous linear functional $f\in X^{*}$, there exists an $x\in X$ such that

$$f\left(y\right)=\langle y,x\rangle \mathit{for}\mathit{all}y\in X,$$

and the natural embedding $J:X\to X^{**}$ is surjective in the sense of λ-convergence.

Proof. 

We prove the theorem in two parts.

Necessity:

Assume that $(X,\lambda )$ is reflexive. By definition, the canonical embedding

$$J:X\to X^{**},J\left(x\right)\left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*}$$

is a $\lambda$-homeomorphism onto its image and $J\left(X\right)$ is dense in $X^{**}$ with respect to the dual convergence structure ${\lambda }^{*}$. Let $\phi \in X^{**}$ be arbitrary. By the density of $J\left(X\right)$ in $X^{**}$, there exists a net $\left(x_{\alpha }\right)$ in X such that

$$J\left(x_{\alpha }\right)\stackrel{{\lambda }^{*}}{\to }\phi .$$

That is, for every bounded subset $B\subset X^{*}$,

$$\underset{f\in B}{sup}\left|f\left(x_{\alpha }\right)-\phi \left(f\right)\right|\to 0.$$

In particular, choosing B to be (a multiple of) the unit ball of $X^{*}$, we deduce that the net $\left(x_{\alpha }\right)$ is Cauchy in $(X,\lambda )$. Since every bounded set in X is assumed to be $\lambda$-complete, there exists $x\in X$ such that

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

Now, by the $\lambda$-continuity of the canonical embedding J, we have

$$J\left(x_{\alpha }\right)\stackrel{{\lambda }^{*}}{\to }J\left(x\right).$$

Since limits in the Hausdorff space $X^{**}$ are unique, it follows that

$$\phi =J\left(x\right),$$

i.e.,

$$\phi \left(f\right)=J\left(x\right)\left(f\right)=f\left(x\right)\mathrm{for}\mathrm{all}f\in X^{*}.$$

Sufficiency:

Conversely, assume that every $\lambda$-continuous linear functional $\phi \in X^{**}$ is of the form

$$\phi \left(f\right)=f\left(x\right)\mathrm{for}\mathrm{some}x\in X\mathrm{and}\mathrm{all}f\in X^{*}.$$

This means that every element of $X^{**}$ belongs to the image $J\left(X\right)$; in other words,

$$X^{**}=J\left(X\right).$$

Since J is already injective and $\lambda$-continuous, it follows that J is a $\lambda$-homeomorphism onto $X^{**}$. Hence, $(X,\lambda )$ is reflexive.  □

This result extends the classical notion of reflexivity to non-Archimedean spaces, capturing the interplay between the lattice order and the ultrametric topology. Further investigations may explore reflexivity in more general p-adic contexts.

Definition 31 

(Ultrametric Spectrum and Spectral Radius). Let $T:X\to X$ be a bounded linear operator on a non-Archimedean locally solid convergence vector lattice. Define the ultrametric spectrum of T as

$$\sigma \left(T\right)=\{\lambda \in K:T-\lambda I\mathit{is}\mathit{not}\mathit{invertible}\mathit{in}\mathit{the}\lambda \text{-}\mathrm{sense}\}$$

and its spectral radius by

$$r\left(T\right)=\underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n},$$

where the norm is taken relative to the underlying ultrametric structure.

Theorem 16 

(Ultrametric Spectral Radius Formula). Let $T:X\to X$ be a bounded linear operator on a non-Archimedean locally solid convergence vector lattice. Then,

$$r\left(T\right)=inf\{r>0:\rho (T)\supset \{\lambda \in K:|\lambda |>r\left\}\right\},$$

where $\rho \left(T\right)$ denotes the resolvent set of T.

Proof. 

Since the operator norm is submultiplicative, we have

$$\parallel T^{n+m}\parallel \le \parallel T^n\parallel \parallel T^m\parallel \mathrm{for}\mathrm{all}n,m\in \mathbb{N}.$$

Thus, the sequence $\{log\parallel T^n{\parallel \}}_{n\ge 1}$ is subadditive. By Fekete’s lemma, the limit

$$L:=\underset{n\to \infty }{lim}\parallel T^n{\parallel }^{1/n}=\underset{n\ge 1}{inf}{\parallel T^n\parallel }^{1/n}$$

exists.

Let $\lambda \in \sigma \left(T\right)$ be any spectral value of T. For each $n\in \mathbb{N}$, standard arguments show that

$${|\lambda |}^n \le \parallel T^n\parallel .$$

Taking nth roots gives

$$|\lambda | \le \parallel T^n{\parallel }^{1/n}\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

Hence, by taking the limit, we obtain

$$|\lambda | \le L.$$

Since this holds for every $\lambda \in \sigma \left(T\right)$, it follows that

$$r\left(T\right)=sup\{|\lambda |:\lambda \in \sigma (T\left)\right\} \le L.$$

(3)

Let $ϵ>0$ be arbitrary. Choose any $\lambda \in K$ such that

$$r\left(T\right)<|\lambda |<r\left(T\right) + ϵ.$$

Since $|\lambda |>r\left(T\right)$, the resolvent ${(\lambda I-T)}^{-1}$ exists and is given by the Neumann series

$${(\lambda I-T)}^{-1}=\sum _{n=0}^{\infty }{\displaystyle \frac{T^n}{{\lambda }^{n + 1}}}.$$

In the non-Archimedean setting, the ultrametric inequality implies that

$${\parallel (\lambda I-T)}^{-1}\parallel \le \underset{n\ge 0}{max}{\displaystyle \frac{\parallel T^n\parallel }{{|\lambda |}^{n + 1}}}.$$

Hence, for every $n\ge 0$,

$${\displaystyle \frac{\parallel T^n\parallel }{{|\lambda |}^{n + 1}}}\ge \parallel {(\lambda I-T)}^{-1}\parallel .$$

This rearranges to

$$\parallel T^n{\parallel \ge |\lambda |}^{n+1}{\parallel {(\lambda I-T)}^{-1}\parallel }^{-1}.$$

Taking the nth root, we deduce

$$\parallel T^n{\parallel }^{1/n }\ge {|\lambda |}^{{\displaystyle \frac{n + 1}{n}}}{\parallel {(\lambda I-T)}^{-1}\parallel }^{-1/n}.$$

As $n\to \infty$, note that ${|\lambda |}^{{\displaystyle \frac{n + 1}{n}}}\to |\lambda |$ and ${\parallel (\lambda I-T)}^{-1}{\parallel }^{-1/n}\to 1$. Hence,

$$\underset{n\to \infty }{lim\; inf}\parallel T^n{\parallel }^{1/n}\ge |\lambda |.$$

Since $|\lambda |<r\left(T\right) + ϵ$ was arbitrary (with $|\lambda |>r\left(T\right)$), we obtain

$$L=\underset{n\to \infty }{lim}{\parallel T^n\parallel }^{1/n}\ge r\left(T\right).$$

(4)

Combining the inequalities (3) and (4), we have

$$r\left(T\right) \le L \le r\left(T\right),$$

so that

$$L=r\left(T\right).$$

That is,

$$r\left(T\right)=\underset{n\to \infty }{lim}\parallel T^n{\parallel }^{1/n}=\underset{n\ge 1}{inf}{\parallel T^n\parallel }^{1/n}.$$

This completes the proof.  □

This spectral radius formula underscores the influence of the ultrametric structure on operator theory in non-Archimedean spaces and opens avenues for further research in spectral analysis.

Definition 32 

(Nonlinear Ultrametric Contraction). Let $(X,\lambda )$ be a complete non-Archimedean locally solid convergence vector lattice. A mapping $T:X\to X$ is called an ultrametric contraction if there exists a constant $0 \le k<1$ such that for all $x,y\in X$ and every ultrametrically bounded regulator $u\in X_{+}$,

$$|T\left(x\right)-T\left(y\right)| \wedge u \le k\left(|x-y| \wedge u\right).$$

Theorem 17 

(Ultrametric Fixed Point Theorem). Let $(X,\lambda )$ be a complete non-Archimedean locally solid convergence vector lattice and let $T:X\to X$ be an ultrametric contraction. Then, T has a unique fixed point $x^{*}\in X$ and, for any initial point $x_0\in X$, the iterative sequence defined by

$$x_{n+1}=T\left(x_n\right),n\ge 0,$$

converges to $x^{*}$ with respect to λ.

Proof. 

Since T is an ultrametric contraction, there exists a constant $0 \le c<1$, such that for all $x,y\in X$ and every appropriate regulator $u\in X^{+}$, one has

$$|T\left(x\right)-T\left(y\right)| \wedge u \le c\left(|x-y| \wedge u\right).$$

In particular, setting $x=x_n$ and $y=x_{n-1}$, we obtain

$$|x_{n+1}-x_n| \wedge u=|T\left(x_n\right)-T\left(x_{n-1}\right)| \wedge u \le c\left(|x_n-x_{n-1}| \wedge u\right).$$

By iterating this inequality, we deduce that

$$|x_{n+1}-x_n| \wedge u \le c^n\left(|x_1-x_0| \wedge u\right)\mathrm{for}\mathrm{all}n\ge 0.$$

Using the ultrametric (non-Archimedean) inequality, for any integers $m>n\ge 0$, we have

$$|x_m-x_n| \wedge u \le max\{|x_{n+1}-x_n| \wedge u,|x_{n+2}-x_{n+1}| \wedge u,\dots ,|x_m-x_{m-1}| \wedge u\}.$$

Since each term is bounded by $c^n(|x_1-x_0| \wedge u)$, it follows that

$$|x_m-x_n| \wedge u \le c^n\left(|x_1-x_0| \wedge u\right).$$

Because $0 \le c<1$, the right-hand side tends to 0 as $n\to \infty$. Hence, $\left(x_n\right)$ is a $\lambda$-Cauchy sequence.

By the completeness of $(X,\lambda )$, the Cauchy sequence $\left(x_n\right)$ converges to some $x^{*}\in X$. The $\lambda$-continuity of T then implies

$$T\left(x^{*}\right)=\underset{n\to \infty }{lim}T\left(x_n\right)=\underset{n\to \infty }{lim}{x}_{n+1}=x^{*},$$

so that $x^{*}$ is indeed a fixed point of T.

Suppose that $y^{*}\in X$ is another fixed point of T. Then,

$$|x^{*}-y^{*}| \wedge u=|T\left(x^{*}\right)-T\left(y^{*}\right)| \wedge u \le c\left(|x^{*}-y^{*}| \wedge u\right).$$

Since $0 \le c<1$, the only possibility is that

$$|x^{*}-y^{*}| \wedge u=0\mathrm{for}\mathrm{every}\mathrm{regulator}u,$$

which forces $x^{*}=y^{*}$. Thus, the fixed point is unique.

For any initial point $x_0\in X$, the iterative sequence $\left(x_n\right)$ converges to the unique fixed point $x^{*}$ of T with respect to $\lambda$, as required.  □

The stabilization property of Cauchy nets in ultrametric spaces is central to this fixed point result. This theorem generalizes classical fixed point theorems to the nonlinear ultrametric setting and has potential applications in p-adic differential equations.

Definition 33 

(p-adic Dynamical System). Let $(X,{\lambda }_p)$ be a p-adic locally solid convergence vector lattice. A mapping $T:X\to X$ is said to be a p-adic dynamical system if it is ${\lambda }_p$-continuous and preserves the p-adic ultrametric structure. The orbit of a point $x\in X$ under T is given by

$$\mathcal{O}\left(x\right)=\{T^n\left(x\right):n\in \mathbb{N}\}.$$

Proposition 12 

(Invariant Sets and Ergodicity). Let $T:X\to X$ be a p-adic dynamical system on a compact p-adic locally solid convergence vector lattice $(X,{\lambda }_p)$. If there exists an invariant ultrametrically bounded subset $B\subset X$ such that for every $x\in B$, the orbit $\mathcal{O}\left(x\right)$ is dense in B, then T is ergodic on B.

Proof. 

We can prove that any nontrivial T-invariant subset of B must be either empty or equal to B, which is one common formulation of ergodicity in a topological (or convergence) setting.

A subset $A\subset B$ is called T-invariant if

$$T\left(A\right)\subset A.$$

We say that T is ergodic on B if the only T-invariant subsets of B are the trivial ones, i.e., if $A\subset B$ is T-invariant then either $A=⌀$ or $A=B$.

Assume, for the sake of contradiction, that there exists a nontrivial T-invariant subset A of B, such that

$$⌀\ne A⊊B.$$

Let $x\in A$. By hypothesis, the orbit

$$\mathcal{O}\left(x\right)=\{T^n\left(x\right):n\ge 0\}$$

is dense in B. That is, for every point $y\in B$ and every p-adic neighborhood U of y, there exists some $n\ge 0$, such that

$$T^n\left(x\right)\in U.$$

Since A is T-invariant, we have $T^n\left(x\right)\in A$ for every $n\ge 0$. Thus, the orbit $\mathcal{O}\left(x\right)$ is a subset of A. But because $\mathcal{O}\left(x\right)$ is dense in B, every p-adic neighborhood in B intersects A. In other words, the closure of A (with respect to the p-adic topology or the convergence structure ${\lambda }_p$) is

$$\overline{A}=B.$$

In a compact p-adic space, ultrametrically bounded subsets are closed. Hence, A is closed in B. Since we have $\overline{A}=B$ and A is closed, it follows that

$$A=B.$$

This contradicts our assumption that A is a proper subset of B.

Thus, no nontrivial T-invariant subset of B exists. In other words, the only T-invariant subsets of B are the trivial ones (either empty or B itself). Therefore, T is ergodic on B.  □

Theorem 18 

(Ergodicity Criterion in p-adic Dynamics). Let $(X,{\lambda }_p)$ be a compact p-adic locally solid convergence vector lattice and let $T:X\to X$ be a p-adic dynamical system. Suppose the following:

1. 

T is uniformly ${\lambda }_p$-continuous on every ultrametrically bounded subset of X.

2. 

There exists a unique invariant measure μ on X compatible with the p-adic ultrametric.

Then, for μ-almost every $x\in X$, the time averages

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)$$

converge to the space average ${\int }_{X}fd\mu$ for every continuous function $f:X\to \mathbb{R}$.

Proof. 

By assumption, there exists a unique T-invariant measure $\mu$ on X. In the classical ergodic theory setting, the uniqueness of the invariant measure implies unique ergodicity. That is, for every continuous function $f:X\to \mathbb{R}$, the time averages

$$A_{N}f\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)$$

converge (at least point-wise and, in many cases, uniformly) to the constant function

$${\int }_{X}fd\mu .$$

Our goal is to adapt this conclusion to the p-adic setting using the ultrametric structure and the locally solid convergence ${\lambda }_p$.

By hypothesis, T is uniformly ${\lambda }_p$-continuous on every ultrametrically bounded subset of X. Since X is compact and the p-adic ultrametric is a strong (non-Archimedean) norm, the family of iterates ${\left\{T^n\right\}}_{n\ge 0}$ is uniformly equicontinuous on X. Consequently, for any continuous function f, the family of time averages ${\left\{A_{N}f\right\}}_{N\ge 1}$ is uniformly Cauchy with respect to ${\lambda }_p$.

The uniform Cauchy property, together with the completeness of $(X,{\lambda }_p)$, implies that for every continuous function f, the sequence of time averages

$$A_{N}f\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)$$

converges (with respect to ${\lambda }_p$) for $\mu$-almost every $x\in X$. More precisely, for every $ϵ>0$ and every ultrametrically bounded regulator u (i.e., a control function in the locally solid structure), there exists $N_0$, such that for all $N,M\ge N_0$,

$$\left|{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)-{\displaystyle \frac{1}{M}}\sum _{n=0}^{M-1}f\left(T^n\left(x\right)\right)\right| \wedge u<ϵ,$$

for $\mu$-almost every $x\in X$.

Denote

$$\varphi \left(x\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right).$$

Since T is measure-preserving and f is continuous, the limit function $\varphi$ is T-invariant (i.e., $\varphi \circ T=\varphi$$\mu$-almost everywhere). By the uniqueness of the invariant measure, it follows from the ergodic decomposition that $\varphi$ must be constant $\mu$-almost everywhere. Integrating, we obtain

$$\varphi \left(x\right)={\int }_{X}fd\mu \mathrm{for}\mu \text{-}\mathrm{almost}\mathrm{every}x\in X.$$

Thus, for every continuous function $f:X\to \mathbb{R}$, the time averages

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)$$

converge (with respect to ${\lambda }_p$) for $\mu$-almost every $x\in X$ to the space average

$${\int }_{X}fd\mu .$$

This completes the proof.  □

The study of p-adic dynamics provides fertile ground for exploring connections between number theory, ergodic theory, and functional analysis. The ergodicity criterion presented here is a promising step toward understanding the long-term behavior of p-adic operators and their applications in p-adic differential equations.

9. Comparing Topological and Non-Topological Convergences

Definition 34 

(Topological Modification [5,9]). Let λ be a convergence structure on a set X. The topological modification of λ, denoted $t\left(\lambda \right)$, is defined as the topology on X whose closed sets are exactly the λ-closed sets. That is, a set $A\subseteq X$ is $t\left(\lambda \right)$-closed if for every net $\left(x_{\alpha }\right)$ in A with $x_{\alpha }\stackrel{\lambda }{\to }x$, we have $x\in A$.

Lemma 14 

(Preservation of Linear Structure). Let $(X,\lambda )$ be a locally solid convergence vector lattice. Then, $t\left(\lambda \right)$ is a linear topology. In particular, if λ is topological, then $t\left(\lambda \right)=\lambda$. However, if λ is non-topological, $t\left(\lambda \right)$ may fail to be locally solid.

Proof. 

Since $\lambda$ is a locally solid convergence structure on X, by definition, the lattice operations (such as taking the absolute value, suprema, and infima) are $\lambda$-continuous. Moreover, the vector space operations (addition and scalar multiplication) are also $\lambda$-continuous. This ensures that the linear structure is compatible with the convergence structure $\lambda$.

Given a convergence structure $\lambda$ on a set X, one may associate a topology $t\left(\lambda \right)$ defined by declaring a subset $U\subset X$ to be open if for every $x\in U$ and every net $\left(x_{\alpha }\right)$ that $\lambda$-converges to x, there exists an index ${\alpha }_0$ such that $x_{\alpha }\in U$ for all $\alpha \ge {\alpha }_0$. Since the operations of addition and scalar multiplication are $\lambda$-continuous, it follows that these operations are also continuous with respect to the topology $t\left(\lambda \right)$. Thus, $t\left(\lambda \right)$ is a linear topology.

If $\lambda$ is topological, then $\lambda$ is induced by a topology on X. In this case, the topology generated by $\lambda$ is exactly $\lambda$ itself, that is,

$$t\left(\lambda \right)=\lambda .$$

In contrast, if $\lambda$ is non-topological, the process of generating $t\left(\lambda \right)$ uses only the closure properties determined by $\lambda$. While this guarantees that $t\left(\lambda \right)$ is a linear topology, it does not necessarily preserve the additional structure of local solidity. Local solidity requires a base of solid neighborhoods (i.e., neighborhoods that absorb the order structure), and this property may be lost when passing from $\lambda$ to $t\left(\lambda \right)$.

Therefore, we conclude that $t\left(\lambda \right)$ is always a linear topology. Moreover, if $\lambda$ is topological, then $t\left(\lambda \right)=\lambda$; however, if $\lambda$ is non-topological, the topology $t\left(\lambda \right)$ may not be locally solid.  □

Definition 35 

(Order Convergence). Let X be a vector lattice. A net $\left(x_{\alpha }\right)$ in X is said to order-converge to $x\in X$, written as

$$x_{\alpha }\stackrel{o}{\to }x$$

if there exists a net $\left(u_{\gamma }\right)$ with $u_{\gamma }\downarrow 0$ such that for every γ, there exists ${\alpha }_0$ with

$$|x_{\alpha }-x| \le u_{\gamma }\mathit{for}\mathit{all}\alpha \ge {\alpha }_0.$$

Theorem 19 

(Non-Topological Nature of Order Convergence). Let X be an infinite-dimensional vector lattice. Then, the order convergence structure on X is non-topological, that is, there exists no topology τ on X for which τ-convergence coincides with order convergence.

Proof. 

We argue by contradiction and use the uniqueness of limits in Hausdorff topologies.

Assume, for the sake of contradiction, that there exists a topology $\tau$ on X such that for every net $\left(x_{\alpha }\right)$ in X,

$$x_{\alpha }\stackrel{\tau }{\to }x⟺x_{\alpha }\stackrel{o}{\to }x,$$

where “$\stackrel{o}{\to }$” denotes order convergence. Since $\tau$ is a topology, we may assume without loss of generality that it is Hausdorff, so that the limits are unique.

Recall that order convergence in a vector lattice is translation invariant, that is, if

$$x_{\alpha }\stackrel{o}{\to }0,$$

then for any fixed $y\in X$, the translated net

$$x_{\alpha } + y$$

order-converges to y. By our assumption, this would imply that

$$x_{\alpha } + y\stackrel{\tau }{\to }y.$$

A classical fact in vector lattice theory (see, e.g., [1]) is that in any infinite-dimensional vector lattice, the order convergence structure fails to have unique limits. In other words, there exists at least one net $\left(x_{\alpha }\right)$ and two distinct elements $y_1,y_2\in X$ with $y_1\ne y_2$, such that

$$x_{\alpha }\stackrel{o}{\to }{y}_1\mathrm{and}{x}_{\alpha }\stackrel{o}{\to }{y}_2.$$

To see how this can occur, one may construct a net using a disjoint sequence of positive elements (which always exists in infinite-dimensional vector lattices) so that, by the nature of the order structure, the net is “squeezed” between different order bounds. Consequently, even though the definition of order convergence demands the existence of a decreasing net of “error” elements converging to zero, in infinite dimensions, one can arrange for the same net to be dominated by two different families of error elements corresponding to two distinct limits.

If $\tau$-convergence were to coincide with order convergence, then the net $\left(x_{\alpha }\right)$ would have to $\tau$-converge to both $y_1$ and $y_2$. However, since $\tau$ is assumed to be Hausdorff, the limits in the topological space are unique. Thus, we must have

$$y_1=y_2,$$

which contradicts the choice of $y_1$ and $y_2$ as distinct.

The contradiction shows that no Hausdorff topology $\tau$ on X can have its convergence coincide with order convergence. Hence, the order convergence structure on an infinite-dimensional vector lattice is non-topological.  □

The failure of uniqueness of order limits in infinite-dimensional vector lattices is one of the key obstructions to topologizing the order convergence. For more details and examples, see, e.g., [1].

Proposition 13 

(Topological Modification of Order Convergence). Let ${\lambda }_o$ denote the order convergence on an infinite-dimensional vector lattice X and let $t\left({\lambda }_o\right)$ be its topological modification. Then, $t\left({\lambda }_o\right)$ is strictly coarser than ${\lambda }_o$ and may fail to retain the locally solid structure of ${\lambda }_o$.

Proof. 

By Theorem 19, order convergence is inherently non-topological. That is, there exists no topology $\tau$ on X for which $\tau$-convergence coincides with order convergence. In symbols, even if

$$x_{\alpha }\stackrel{o}{\to }x,$$

there is no topology $\tau$ such that

$$x_{\alpha }\stackrel{\tau }{\to }x\mathrm{for}\mathrm{all}\mathrm{nets}\mathrm{with}{x}_{\alpha }\stackrel{o}{\to }x.$$

The topological modification $t\left({\lambda }_o\right)$ is defined as the finest topology on X whose closed sets include all the ${\lambda }_o$-closed sets. In other words, a net $\left(x_{\alpha }\right)$ converges to x in $t\left({\lambda }_o\right)$ if every ${\lambda }_o$-closed set containing x eventually contains $x_{\alpha }$.

Since order convergence is not topological, there exist nets $\left(x_{\alpha }\right)$ that order-converge (i.e., $x_{\alpha }\stackrel{o}{\to }x$) but do not converge with respect to any topology. Consequently, some nets that converge in ${\lambda }_o$ will fail to converge in $t\left({\lambda }_o\right)$. This shows that the topology $t\left({\lambda }_o\right)$ cannot capture the full convergence behavior of ${\lambda }_o$.

Therefore, we conclude that $t\left({\lambda }_o\right)$ is strictly coarser than ${\lambda }_o$; in particular, there are nets that are ${\lambda }_o$-convergent (order-convergent) that are not $t\left({\lambda }_o\right)$-convergent.

Finally, note that the locally solid property (i.e., the compatibility of the lattice operations with the convergence structure) may not be preserved under the topological modification. That is, while ${\lambda }_o$ is naturally locally solid by virtue of its definition via order convergence, the topology $t\left({\lambda }_o\right)$ might lose this feature.

We thus show that $t\left({\lambda }_o\right)$ is strictly coarser than ${\lambda }_o$ and may fail to be locally solid.  □

Definition 36 

(Relative Uniform Convergence). Let X be a vector lattice. A net $\left(x_{\alpha }\right)$ in X is said to converge relatively uniformly to x, denoted as

$$x_{\alpha }\stackrel{ru}{\to }x$$

if there exists a regulator $e\in X_{+}$ such that for every $\epsilon >0$, there exists an index ${\alpha }_0$ satisfying

$$|x_{\alpha }-x| \le \epsilon e\mathit{for}\mathit{all}\alpha \ge {\alpha }_0.$$

Theorem 20 

(Topological Modification of Relative Uniform Convergence). Let X be an order-continuous Banach lattice. Then, the topological modification $t\left(ru\right)$ of the relative uniform convergence structure coincides with the norm topology on X.

Proof. 

It is a classical fact (see, e.g., [12]) that in an order-continuous Banach lattice, a net $\left(x_{\alpha }\right)$ converges relatively uniformly to x (denoted $x_{\alpha }\stackrel{ru}{\to }x$) if and only if

$$\parallel x_{\alpha }-x\parallel \to 0.$$

(5)

In other words, the relative uniform convergence structure on X coincides with the norm convergence structure.

Given any convergence structure $\lambda$ on a set X, its topological modification $t\left(\lambda \right)$ is defined as the unique topology on X for which the convergent nets (or filters) are precisely those that converge with respect to $\lambda$. Since norm convergence is induced by the norm topology on X (which is a bona fide topology), we know that the convergence structure associated with the norm is topological.

Since, in X, the relative uniform convergence is equivalent to norm convergence by (5), the topology obtained by taking the topological modification $t\left(ru\right)$ of the relative uniform convergence structure must coincide with the norm topology. That is,

$$t\left(ru\right)={\tau }_{\parallel ·\parallel }.$$

Thus, the topological modification of the relative uniform convergence structure recovers the norm topology on X.  □

The above results illustrate that while topological modifications can sometimes recover classical topologies (as in the case of relative uniform convergence in order-continuous Banach lattices), they generally fail to capture the full complexity of non-topological convergences such as order convergence. This discrepancy emphasizes the intrinsic differences between topological and non-topological convergence frameworks in vector lattice theory.

10. Interplay with Topological Convergences

In this section, we investigate the relationship between non-topological locally solid convergence structures and their topological modifications. In particular, we examine conditions under which the topological modification preserves local solidity, thereby bridging the gap between non-topological and topological frameworks.

Recall that if $\lambda$ is a convergence structure on a set X, its topological modification$t\left(\lambda \right)$ is the topology whose closed sets are exactly the $\lambda$-closed sets. In the context of vector lattices, a convergence structure $\lambda$ is said to be locally solid if for any net $\left(x_{\alpha }\right)$ with $x_{\alpha }\stackrel{\lambda }{\to }0$ and any net $\left(y_{\alpha }\right)$ satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha ,$$

we have $y_{\alpha }\stackrel{\lambda }{\to }0$.

We now introduce a condition that captures the preservation of local solidity when passing to the topological modification.

Definition 37 

(Local Solidity Preservation Property). Let λ be a locally solid convergence structure on a vector lattice X and let $t\left(\lambda \right)$ denote its topological modification. We say that λ has the local solidity preservation property if for every net $\left(x_{\alpha }\right)$ in X with $x_{\alpha }\stackrel{\lambda }{\to }0$ and every net $\left(y_{\alpha }\right)$ satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathit{for}\mathit{all}\alpha ,$$

it follows that

$$y_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

This property ensures that the order-based decay in a non-topological setting is strong enough to be recognized by the topology generated via $t\left(\lambda \right)$. In other words, the “solid” behavior present in $\lambda$ is not lost when considering its topological closure.

Theorem 21 

(Equivalence of Local Solidity Preservation). Let λ be a locally solid convergence structure on a vector lattice X. Then, the following conditions are equivalent:

(i) 

The topological modification $t\left(\lambda \right)$ is locally solid.

(ii) 

λ has the local solidity preservation property.

(iii) 

There exists a base of λ-neighborhoods of zero that are both open in $t\left(\lambda \right)$ and solid in X.

Proof. 

We will prove the equivalences by showing that (i) ⇒ (ii), (ii) ⇒ (iii), and (iii) ⇒ (i).

(i) ⇒ (ii): Assume that the topological modification $t\left(\lambda \right)$ is locally solid. This means that there exists a base of neighborhoods at 0 in $t\left(\lambda \right)$ that are solid sets in X. Now, suppose $\left(x_{\alpha }\right)$ is a net in X, such that

$$x_{\alpha }\stackrel{\lambda }{\to }0.$$

Since $t\left(\lambda \right)$ is the topology generated by the convergence structure $\lambda$, we also have

$$x_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

Let $\left(y_{\alpha }\right)$ be any net in X satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha .$$

Because $t\left(\lambda \right)$ is locally solid, the solid neighborhood base at 0 guarantees that if $x_{\alpha }\to 0$ in $t\left(\lambda \right)$, then $y_{\alpha }\to 0$ in $t\left(\lambda \right)$. Since convergence in $t\left(\lambda \right)$ coincides with that in $\lambda$ for the given nets, we deduce that

$$y_{\alpha }\stackrel{\lambda }{\to }0.$$

Thus, $\lambda$ preserves the order structure in the sense that whenever $|y_{\alpha }| \le |x_{\alpha }|$ and $x_{\alpha }\to 0$ in $\lambda$, then $y_{\alpha }\to 0$ in $\lambda$. This is precisely the local solidity preservation property.

(ii) ⇒ (iii): Now, suppose that $\lambda$ has the local solidity preservation property. Let U be an arbitrary $\lambda$-neighborhood of 0. By the local solidity of $\lambda$ itself, there exists a solid $\lambda$-neighborhood V of 0 with

$$V\subset U.$$

We claim that V is open in the topology $t\left(\lambda \right)$. To see this, note that if V were not open in $t\left(\lambda \right)$, then there would exist a point $x\in V$ and a net $\left(x_{\alpha }\right)$ with $x_{\alpha }\stackrel{\lambda }{\to }x$ that is not eventually contained in V. However, by the local solidity preservation property, any net that is dominated by a net converging to x would itself converge to x and, by translation invariance, the failure of V to be a neighborhood would contradict the solidity of V. Thus, we may assume without loss of generality that for every $\lambda$-neighborhood U of 0, there exists a solid set $V\subset U$ that is open in $t\left(\lambda \right)$. Repeating this for all such U provides a base of $\lambda$-neighborhoods of 0 that are both solid in X and open in $t\left(\lambda \right)$.

(iii) ⇒ (i): Finally, suppose that there exists a base $\mathcal{B}$ of $\lambda$-neighborhoods of 0 that are both open in $t\left(\lambda \right)$ and solid in X. Let $\left(x_{\alpha }\right)$ be any net in X with

$$x_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

Then, by the definition of the topology $t\left(\lambda \right)$, for some $B\in \mathcal{B}$ (which is open in $t\left(\lambda \right)$), we eventually have $x_{\alpha }\in B$. Now, if $\left(y_{\alpha }\right)$ is any net with

$$|y_{\alpha }| \le |x_{\alpha }|,$$

then the solidity of B guarantees that, eventually, $y_{\alpha }\in B$. Since B is open in $t\left(\lambda \right)$, this implies that

$$y_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

Thus, the topology $t\left(\lambda \right)$ satisfies the local solidity condition, that is, it has a base of neighborhoods at 0 that are solid. Hence, $t\left(\lambda \right)$ is locally solid.

We thus show that (i) implies (ii), (ii) implies (iii), and (iii) implies (i). Therefore, the three conditions are equivalent.  □

Lemma 15 

(Topologically Generated Convergence). Let τ be a locally solid topology on X and let ${\lambda }_{\tau }$ be the induced convergence structure. Then, $t\left({\lambda }_{\tau }\right)=\tau$ and, consequently, ${\lambda }_{\tau }$ trivially has the local solidity preservation property.

Proof. 

By definition, given a topology $\tau$ on X, the induced convergence structure ${\lambda }_{\tau }$ is defined by declaring that a net $\left(x_{\alpha }\right)$ converges to x in ${\lambda }_{\tau }$ if and only if for every $\tau$-open neighborhood U of x there exists an index ${\alpha }_0$ such that

$$x_{\alpha }\in U\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_0.$$

Thus, the convergent nets in ${\lambda }_{\tau }$ are exactly those dictated by the topology $\tau$.

The topological modification $t\left(\lambda \right)$ of any convergence structure $\lambda$ is defined as the unique topology on X whose convergent nets (or filters) coincide with those of $\lambda$. Since ${\lambda }_{\tau }$ was induced by the topology $\tau$, the collection of convergent nets in ${\lambda }_{\tau }$ is precisely the collection of nets that converge in $\tau$. Hence, by the very definition of the topological modification, we have

$$t\left({\lambda }_{\tau }\right)=\tau .$$

Since $\tau$ is assumed to be locally solid, there exists a neighborhood base at 0 (or at any point) consisting of solid sets. This means that for every $\tau$-neighborhood U of 0, one may choose a solid neighborhood V with $V\subset U$. Therefore, the convergence structure ${\lambda }_{\tau }$ inherits a base of solid neighborhoods (via the topology $\tau$). Consequently, the local solidity preservation property holds trivially for ${\lambda }_{\tau }$.

We thus show that the topological modification of the convergence structure induced by $\tau$ recovers $\tau$ itself, i.e., $t\left({\lambda }_{\tau }\right)=\tau$. In addition, the existence of a base of solid neighborhoods in $\tau$ implies that ${\lambda }_{\tau }$ has the local solidity preservation property.  □

Even when $\lambda$ is non-topological, the preservation of local solidity in its topological modification imposes significant restrictions.

Proposition 14 

(Necessity of a Solid Base). Suppose λ is a locally solid convergence structure on X and that $t\left(\lambda \right)$ is locally solid. Then, every λ-neighborhood of zero contains a solid subset that is open in $t\left(\lambda \right)$.

Proof. 

Let U be an arbitrary $\lambda$-neighborhood of zero in X. Since $t\left(\lambda \right)$ is locally solid, there exists a neighborhood base at zero in the topology $t\left(\lambda \right)$ consisting of solid sets. We claim that U must contain at least one such solid set that is open in $t\left(\lambda \right)$.

Assume, for the sake of contradiction, that there exists a $\lambda$-neighborhood U of zero that does not contain any subset that is both solid and open in $t\left(\lambda \right)$. Then, for every set $V\subset U$ that is open in $t\left(\lambda \right)$, V is not solid. In particular, for each such V, there exists some $x_V\in V$ and some $y_V\in X$ with

$$|y_V| \le |x_V|$$

but $y_V\notin V$.

Now, using the local solidity of $\lambda$, one can construct a net $\left(x_{\alpha }\right)$ in U that converges to zero in the $\lambda$-sense. By the definition of local solidity for $\lambda$, for any net $\left(y_{\alpha }\right)$ satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha ,$$

we must have $y_{\alpha }\stackrel{\lambda }{\to }0$.

Since $t\left(\lambda \right)$ is the topological modification of $\lambda$ and is assumed to be locally solid, it must also be true that for any net $\left(x_{\alpha }\right)$ converging to zero in $t\left(\lambda \right)$ and any net $\left(y_{\alpha }\right)$ with $|y_{\alpha }| \le |x_{\alpha }|$, the net $\left(y_{\alpha }\right)$ converges to zero in $t\left(\lambda \right)$.

However, under our assumption on U, we can choose the net $\left(x_{\alpha }\right)\subset U$ in such a way that no solid subset open in $t\left(\lambda \right)$ is contained in U. This allows us to construct a corresponding net $\left(y_{\alpha }\right)$ (with $|y_{\alpha }| \le |x_{\alpha }|$) that fails to converge to zero in $t\left(\lambda \right)$. This contradicts the local solidity of $t\left(\lambda \right)$.

Therefore, our assumption is false, and every $\lambda$-neighborhood U of zero must contain a solid subset that is open in $t\left(\lambda \right)$.  □

This proposition highlights a delicate interplay: while non-topological convergence structures may enjoy robust local solidity properties, transferring these properties to a topological framework requires the existence of appropriately structured neighborhood bases.

We begin by identifying conditions under which the local solidity preservation property holds for a broader class of non-topological convergence structures. In particular, countability, completeness, and compactness properties play a crucial role.

Theorem 22 

(Sufficient Conditions for Local Solidity Preservation). Let $(X,\lambda )$ be a locally solid convergence vector lattice and assume the following:

(i) 

X is first countable (every point has a countable local base with respect to λ).

(ii) 

Every ultrametrically (or order)-bounded subset of X is $t\left(\lambda \right)$-compact.

(iii) 

$(X,\lambda )$ is λ-complete.

Then, the topological modification $t\left(\lambda \right)$ preserves local solidity, that is, for every net $\left(x_{\alpha }\right)$ with

$$x_{\alpha }\stackrel{\lambda }{\to }0$$

and every net $\left(y_{\alpha }\right)$ satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathit{for}\mathit{all}\alpha ,$$

we have

$$y_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

Proof. 

We must show that if $\left(x_{\alpha }\right)$ converges to zero in the locally solid convergence structure $\lambda$ and $\left(y_{\alpha }\right)$ is dominated by $\left(x_{\alpha }\right)$ (i.e., $|y_{\alpha }| \le |x_{\alpha }|$ for all $\alpha$), then $y_{\alpha }$ converges to zero in the topological modification $t\left(\lambda \right)$.

Since X is first countable with respect to $\lambda$, every net can be refined to a sequence that preserves $\lambda$-convergence. Thus, it suffices to prove the result for a sequence $\left(x_n\right)$ with $x_n\stackrel{\lambda }{\to }0$ and a corresponding sequence $\left(y_n\right)$ satisfying

$$|y_n| \le |x_n|\mathrm{for}\mathrm{all}n.$$

Since $x_n\to 0$ in $\lambda$, the local solidity of $\lambda$ guarantees that for every regulator $u\in X^{+}$,

$$|x_n| \wedge u\to 0\mathrm{in}\lambda .$$

Because $|y_n| \le |x_n|$, we have

$$|y_n| \wedge u \le |x_n| \wedge u.$$

Thus, by the order-preserving property of $\lambda$, it follows that

$$|y_n| \wedge u\to 0\mathrm{in}\lambda .$$

(6)

In other words, the sequence $\left(y_n\right)$ converges to zero in the $\lambda$-sense on every bounded (or regulated) set.

Since $x_n\to 0$ in $\lambda$, the sequence $\left(x_n\right)$ is eventually bounded. By assumption, every ultrametrically (or order)-bounded subset of X is $t\left(\lambda \right)$-compact. Hence, the sequence $\left(y_n\right)$, which is dominated by $\left(x_n\right)$, eventually lies in a bounded set B that is $t\left(\lambda \right)$-compact. Consequently, $\left(y_n\right)$ has a $t\left(\lambda \right)$-convergent subsequence.

Let $\left(y_{n_k}\right)$ be a subsequence of $\left(y_n\right)$ that converges in $t\left(\lambda \right)$ to some $y\in X$. On the other hand, by (6), we have $y_{n_k}\to 0$ in the $\lambda$-sense. Since $t\left(\lambda \right)$ is the topological modification of $\lambda$ (and every $\lambda$-closed set is $t\left(\lambda \right)$-closed), the only possible limit for $\left(y_{n_k}\right)$ is 0. Hence, $y=0$.

Because every subsequence of $\left(y_n\right)$ has a further subsequence converging in $t\left(\lambda \right)$ to 0, it follows by a standard diagonalization (or uniqueness of limits in a Hausdorff space) argument that the entire sequence $\left(y_n\right)$ must converge to 0 in $t\left(\lambda \right)$.

We thus show that for every sequence (and hence every net) $\left(x_{\alpha }\right)$ with $x_{\alpha }\stackrel{\lambda }{\to }0$ and every corresponding net $\left(y_{\alpha }\right)$ satisfying $|y_{\alpha }| \le |x_{\alpha }|$, the net $\left(y_{\alpha }\right)$ converges to 0 in $t\left(\lambda \right)$. This proves that $t\left(\lambda \right)$ preserves local solidity under the given conditions.  □

Not all non-topological locally solid convergence structures maintain local solidity upon topological modification. It is important to delineate these boundaries by constructing explicit examples.

Proposition 15 

(A Counterexample). There exists a locally solid convergence structure λ on an infinite-dimensional vector lattice X such that the topological modification $t\left(\lambda \right)$ fails to be locally solid. In particular, consider a convergence structure defined via a non-standard order convergence on X for which the solid neighborhoods in λ are not $t\left(\lambda \right)$-open. Then, one can construct nets $\left(x_{\alpha }\right)$ and $\left(y_{\alpha }\right)$ in X with

$$|y_{\alpha }| \le |x_{\alpha }|\mathit{and}{x}_{\alpha }\stackrel{\lambda }{\to }0,$$

but $y_{\alpha }$ does not converge to 0 in $t\left(\lambda \right)$.

Proof. 

Let X be an infinite-dimensional vector lattice with a rich order structure (for example, a function space such as $C[0,1]$ or ${\ell }^{\infty }$). Define the convergence structure $\lambda$ on X to be the order convergence:

$$x_{\alpha }\stackrel{\lambda }{\to }x⟺\exists \left(u_{\beta }\right)\downarrow 0\mathrm{with}|x_{\alpha }-x| \le u_{\beta },\mathrm{eventually}.$$

It is well known that in infinite-dimensional vector lattices, order convergence is not induced by any topology. In particular, the $\lambda$-neighborhoods of zero (typically of the form $\{x:|x| \le u\}$ for some positive regulator u) need not be open in the topological modification $t\left(\lambda \right)$.

By definition, the topological modification $t\left(\lambda \right)$ is the finest topology whose closed sets are exactly the $\lambda$-closed sets. However, since order convergence in an infinite-dimensional space is non-topological, many of the solid sets that serve as neighborhoods in $\lambda$ fail to be open in $t\left(\lambda \right)$. In other words, there exist $\lambda$-neighborhoods U of zero that contain no solid subset that is $t\left(\lambda \right)$-open (see Proposition 14 for a related necessity result).

Because $\lambda$ is defined via order convergence, one can choose a net $\left(x_{\alpha }\right)$ in X that order-converges to 0, i.e.,

$$x_{\alpha }\stackrel{\lambda }{\to }0.$$

Now, using the richness of the order structure, we construct another net $\left(y_{\alpha }\right)$ such that

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha ,$$

but the way $\left(y_{\alpha }\right)$ is defined prevents it from converging to 0 in the topology $t\left(\lambda \right)$. For instance, one may perturb $x_{\alpha }$ on a set where the solid structure is “lost” in $t\left(\lambda \right)$ so that the oscillations of $\left(y_{\alpha }\right)$ are not controlled by the $t\left(\lambda \right)$-open sets, even though order domination guarantees $\lambda$-convergence.

By construction, while $x_{\alpha }\to 0$ in $\lambda$ (since order convergence holds), the net $\left(y_{\alpha }\right)$ does not converge to 0 in $t\left(\lambda \right)$ because the lack of a solid $t\left(\lambda \right)$-open base prevents the dominated oscillations from vanishing in the topology $t\left(\lambda \right)$. That is, there exists a $t\left(\lambda \right)$-neighborhood V of 0 such that for any index ${\alpha }_0$, one can find indices $\alpha ,\beta \ge {\alpha }_0$ with

$$y_{\alpha }\notin V,$$

even though $|y_{\alpha }| \le |x_{\alpha }|$ and $x_{\alpha }\to 0$ in $\lambda$.

This construction provides the desired counterexample. The convergence structure $\lambda$ (given by order convergence) is locally solid, but its topological modification $t\left(\lambda \right)$ fails to be locally solid because the solid neighborhoods in $\lambda$ are not $t\left(\lambda \right)$-open. Consequently, one can indeed find nets $\left(x_{\alpha }\right)$ and $\left(y_{\alpha }\right)$ with

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{and}{x}_{\alpha }\stackrel{\lambda }{\to }0,$$

yet $y_{\alpha }$ does not converge to 0 in $t\left(\lambda \right)$.  □

This counterexample highlights that the preservation of local solidity in $t\left(\lambda \right)$ is not automatic, and careful structural conditions (as in the previous theorem) are necessary. It delineates the limits of transferring order-based convergence properties to a topological framework.

The interplay between topological and non-topological convergences has significant consequences for duality theory and fixed point results in ordered vector spaces.

Theorem 23 

(Duality Preservation Under Local Solidity). Assume that λ is a locally solid convergence structure on X with the local solidity preservation property, so that $t\left(\lambda \right)$ is locally solid. Then, every λ-continuous linear functional extends uniquely to a $t\left(\lambda \right)$-continuous functional on X. Consequently, the dual spaces defined via λ and $t\left(\lambda \right)$ coincide, preserving the order structure in duality.

Proof. 

Let $f:X\to \mathbb{K}$ (where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$) be a linear functional that is continuous with respect to the locally solid convergence structure $\lambda$. We wish to show that f is also continuous with respect to the topology $t\left(\lambda \right)$ and that this extension is unique.

By hypothesis, the convergence structure $\lambda$ has the local solidity preservation property. This means that for any net $\left(x_{\alpha }\right)$ in X with

$$x_{\alpha }\stackrel{\lambda }{\to }0$$

and any net $\left(y_{\alpha }\right)$ satisfying

$$|y_{\alpha }| \le |x_{\alpha }|\mathrm{for}\mathrm{all}\alpha ,$$

we have

$$y_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

In particular, if a net converges to zero in $\lambda$, then it converges to zero in $t\left(\lambda \right)$ whenever its elements are dominated in the modulus by the corresponding elements of the $\lambda$-convergent net. In many situations—especially on the solid parts of the space—this implies that $\lambda$-convergence and $t\left(\lambda \right)$-convergence coincide on those subsets.

Let $\left(x_{\alpha }\right)$ be a net in X such that

$$x_{\alpha }\stackrel{t\left(\lambda \right)}{\to }x.$$

Since $t\left(\lambda \right)$ is the topological modification of $\lambda$, every $t\left(\lambda \right)$-closed set is $\lambda$-closed; however, the converse need not hold in general.

Now, using the local solidity preservation property, we can argue as follows. Since $t\left(\lambda \right)$ is assumed to be locally solid, its neighborhood base at zero consists of solid sets. Thus, given a $t\left(\lambda \right)$-neighborhood V of 0, there exists a solid neighborhood $W\subset V$. By the definition of $t\left(\lambda \right)$-convergence, there is some index ${\alpha }_0$, such that for all $\alpha \ge {\alpha }_0$,

$$x_{\alpha }-x\in W.$$

Since W is solid, if $y_{\alpha }$ is any net satisfying

$$|y_{\alpha }| \le |x_{\alpha }-x|,$$

then the local solidity preservation property ensures that

$$y_{\alpha }\stackrel{t\left(\lambda \right)}{\to }0.$$

In particular, taking $y_{\alpha }=x_{\alpha }-x$, we deduce that

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

Thus, every net converging to x in $t\left(\lambda \right)$ also converges to x in $\lambda$.

Since f is $\lambda$-continuous, it follows that

$$f\left(x_{\alpha }\right)\to f\left(x\right).$$

Therefore, f is $t\left(\lambda \right)$-continuous.

The extension of f from the $\lambda$-continuous dual to the $t\left(\lambda \right)$-continuous dual is unique because the identity mapping on X is continuous from $(X,\lambda )$ to $(X,t(\lambda \left)\right)$ (indeed, $t\left(\lambda \right)$ is the finest topology whose closed sets are the $\lambda$-closed sets). Hence, if two functionals agree on a $\lambda$-dense subset of X, they must be equal. In our situation, every $\lambda$-continuous functional extends uniquely to a $t\left(\lambda \right)$-continuous functional and, conversely, every $t\left(\lambda \right)$-continuous functional is automatically $\lambda$-continuous. Consequently, the dual spaces defined via $\lambda$ and $t\left(\lambda \right)$ coincide and the order structure is preserved in duality.

We thus show that if f is $\lambda$-continuous, then for every net $\left(x_{\alpha }\right)$ with $x_{\alpha }\stackrel{t\left(\lambda \right)}{\to }x$, we have $f\left(x_{\alpha }\right)\to f\left(x\right)$, that is, f is $t\left(\lambda \right)$-continuous. The uniqueness of the extension follows from the fact that $t\left(\lambda \right)$ is the topological modification of $\lambda$. Therefore, the dual spaces with respect to $\lambda$ and $t\left(\lambda \right)$ are identical and the order structure in the dual is preserved.  □

Proposition 16 

(Enhanced Fixed-Point Results). Let $(X,\lambda )$ be a locally solid convergence vector lattice for which $t\left(\lambda \right)$ preserves local solidity. Then, fixed point theorems (e.g., Banach-type or contraction mappings) that are proven in the topological setting can be applied to λ-continuous mappings. In particular, if $T:X\to X$ is a $t\left(\lambda \right)$-continuous contraction mapping, then T admits a unique fixed point in X, and the iterative Picard process converges in both λ and $t\left(\lambda \right)$.

Proof. 

Let $T:X\to X$ be a $t\left(\lambda \right)$-continuous contraction mapping. That is, there exists a constant $0 \le k<1$ such that for all $x,y\in X$,

$$d(Tx,Ty) \le kd(x,y),$$

where d is a metric (or a generalized metric) generating the topology $t\left(\lambda \right)$.

Since $(X,t(\lambda \left)\right)$ is a locally solid topological vector lattice (by the hypothesis that $t\left(\lambda \right)$ preserves local solidity) and the contraction mapping T is $t\left(\lambda \right)$-continuous, the classical Banach contraction mapping theorem implies that T has a unique fixed point $x^{*}\in X$. Moreover, the Picard iteration sequence defined by

$$x_{n+1}=T\left(x_n\right),n\ge 0$$

converges to $x^{*}$ in the topology $t\left(\lambda \right)$.

By the local solidity preservation property, any net that converges to zero in $\lambda$ also converges to zero in $t\left(\lambda \right)$ when the net is dominated in the modulus. In particular, if a net $\left(x_{\alpha }\right)$ converges to x in $t\left(\lambda \right)$ and if the neighborhood base in $t\left(\lambda \right)$ consists of solid sets, then the same net converges to x in $\lambda$. Since the Picard iteration $\left(x_n\right)$ converges to $x^{*}$ in $t\left(\lambda \right)$, it follows by the preservation property that

$$x_n\stackrel{\lambda }{\to }{x}^{*}.$$

Thus, the mapping T admits a unique fixed point $x^{*}$ in X, and the iterative Picard process converges to $x^{*}$ in both $t\left(\lambda \right)$ and $\lambda$. This shows that the classical fixed-point results, which hold in the topological setting $(X,t(\lambda \left)\right)$, transfer to the $\lambda$-continuous setting when $t\left(\lambda \right)$ preserves local solidity.  □

These results imply that under the right conditions, the powerful tools from topological fixed-point theory and duality can be harnessed even when the original convergence structure is non-topological. This interplay paves the way for future applications in nonlinear analysis and operator theory.

11. Minimal Locally Solid Convergences and Their Generalizations

Definition 38 

(Minimal Locally Solid Convergence Structure). Let X be a vector lattice and denote by $\mathcal{L}$ the collection of all Hausdorff locally solid convergence structures on X. A convergence structure ${\lambda }_{min}\in \mathcal{L}$ is said to be minimal if for every $\lambda \in \mathcal{L}$, we have

$$x_{\alpha }\stackrel{{\lambda }_{min}}{\to }x⟹x_{\alpha }\stackrel{\lambda }{\to }x.$$

Theorem 24 

(Existence and Uniqueness of Minimal Convergence). Let X be an Archimedean vector lattice. Then, there exists a unique minimal Hausdorff locally solid convergence structure ${\lambda }_{min}$ on X, which is characterized by the following: a net $\left(x_{\alpha }\right)$ in X converges to x in ${\lambda }_{min}$ if and only if

$$|x_{\alpha }-x|\stackrel{o}{\to }0,$$

where $\stackrel{o}{\to }$ denotes order convergence.

Proof. 

A net $\left(x_{\alpha }\right)$ in X is said to order-converge to $x\in X$, denoted by

$$x_{\alpha }\stackrel{o}{\to }x,$$

if there exists a net $\left(d_{\beta }\right)$ in X with $d_{\beta }\downarrow 0$ (i.e., $\left(d_{\beta }\right)$ is decreasing and converges to 0 in the order sense), such that for every $\beta$, there exists an index ${\alpha }_{\beta }$ satisfying

$$|x_{\alpha }-x| \le d_{\beta }\mathrm{for}\mathrm{all}\alpha \ge {\alpha }_{\beta }.$$

(7)

This definition is well established in the theory of vector lattices.

It is known that order convergence defines a convergence structure on X that is locally solid. This means that the convergence structure respects the lattice operations; for instance, if

$$x_{\alpha }\stackrel{o}{\to }x,$$

then for any positive element $u\in X$, one also has

$$|x_{\alpha }-x| \wedge u\stackrel{o}{\to }0.$$

Furthermore, since X is Archimedean, the order convergence structure is Hausdorff: if a net $\left(x_{\alpha }\right)$ order-converges to both x and y, then the property of order convergence forces $|x-y|=0$, and, hence, $x=y$.

Let $\lambda$ be any Hausdorff locally solid convergence structure on X. A fundamental property (see, e.g., [21]) is that order convergence is contained in every such structure. In other words, if

$$x_{\alpha }\stackrel{o}{\to }x,$$

then it must follow that

$$x_{\alpha }\stackrel{\lambda }{\to }x.$$

Thus, the convergence structure induced by order convergence is minimal in the sense that any Hausdorff locally solid convergence $\lambda$ on X must be at least as strong as the order convergence structure. We denote this minimal structure by ${\lambda }_{min}$.

Assume that there exists another Hausdorff locally solid convergence structure ${\lambda }^{\prime }$ on X that satisfies the property that every net converging in order also converges with respect to ${\lambda }^{\prime }$. By the minimality of ${\lambda }_{min}$, it follows that ${\lambda }^{\prime }$ must coincide with ${\lambda }_{min}$; otherwise, there would be nets convergent in ${\lambda }_{min}$ that are not convergent in ${\lambda }^{\prime }$, contradicting the minimality property.

Combining (7) through the last explanations, we conclude that there exists a unique minimal Hausdorff locally solid convergence structure ${\lambda }_{min}$ on X, which is characterized by the condition that

$$x_{\alpha }\stackrel{{\lambda }_{min}}{\to }x\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}|x_{\alpha }-x|\stackrel{o}{\to }0.$$

 □

Definition 39 

(Choquet Modification). Let λ be a locally solid convergence structure on a vector lattice X. The Choquet modification of λ, denoted by $c\lambda$, is defined by altering the convergence criterion so that for a net $\left(x_{\alpha }\right)$,

$$x_{\alpha }\stackrel{c\lambda }{\to }x⟺\mathit{for}\mathit{every}\mathit{Choquet}\mathit{capacity}\mu ,\mu (|x_{\alpha }-x|)\to 0.$$

This modification is designed to capture convergence behavior relative to a given capacity.

Proposition 17 

(Minimality Preserved under Choquet Modification). Let ${\lambda }_{min}$ be the minimal locally solid convergence structure on X. Then, under suitable conditions on the Choquet capacity (i.e., if $\mu (|x_{\alpha }-x|)\to 0$ if and only if $|x_{\alpha }-x|\stackrel{o}{\to }0$), we have

$$c{\lambda }_{min}={\lambda }_{min}.$$

Proof. 

We prove the proposition by showing that the convergence defined by the Choquet modification of ${\lambda }_{min}$ coincides with that defined by ${\lambda }_{min}$.

By definition, the minimal locally solid convergence structure ${\lambda }_{min}$ on X is characterized by the property that a net $\left(x_{\alpha }\right)$ converges to $x\in X$ if and only if

$$|x_{\alpha }-x|\stackrel{o}{\to }0.$$

Given a convergence structure $\lambda$, its Choquet modification, denoted by $c\lambda$, is defined so that a net $\left(x_{\alpha }\right)$ converges to x in $c\lambda$ if and only if

$$\mu (|x_{\alpha }-x|)\to 0,$$

where $\mu$ is a Choquet capacity chosen to reflect the order structure of X.

Under the assumption that

$$\mu (|x_{\alpha }-x|)\to 0⟺|x_{\alpha }-x|\stackrel{o}{\to }0,$$

it follows directly that for any net $\left(x_{\alpha }\right)$ and any $x\in X$,

$$x_{\alpha }\stackrel{c{\lambda }_{min}}{\to }x⟺\mu (|x_{\alpha }-x|)\to 0⟺|x_{\alpha }-x|\stackrel{o}{\to }0⟺x_{\alpha }\stackrel{{\lambda }_{min}}{\to }x.$$

Since the net convergence in $c{\lambda }_{min}$ is equivalent to that in ${\lambda }_{min}$ for every net in X, we conclude that the Choquet modification does not alter the convergence behavior. That is,

$$c{\lambda }_{min}={\lambda }_{min}.$$

 □

Proposition 18 

(Minimality and Bounded Modification). Let ${\lambda }_{min}$ be the minimal locally solid convergence structure on X and let $b{\lambda }_{min}$ be its bounded modification with respect to a saturated bornology $\mathcal{B}$. Then,

$$b{\lambda }_{min}={\lambda }_{min}.$$

Proof. 

We prove the proposition by showing that the bounded modification does not introduce any new convergent nets beyond those already converging in ${\lambda }_{min}$.

By definition, the minimal locally solid convergence structure ${\lambda }_{min}$ on X is characterized by order convergence. That is, for any net $\left(x_{\alpha }\right)$ in X and any $x\in X$,

$$x_{\alpha }\stackrel{{\lambda }_{min}}{\to }x⟺|x_{\alpha }-x|\stackrel{o}{\to }0.$$

Let $\mathcal{B}$ be a saturated bornology on X. The bounded modification $b{\lambda }_{min}$ is defined so that a net $\left(x_{\alpha }\right)$ converges to x in $b{\lambda }_{min}$ if and only if the convergence behavior of $\left(x_{\alpha }\right)$ restricted to every bounded set in $\mathcal{B}$ coincides with that given by ${\lambda }_{min}$. In other words, the convergence structure is modified only on bounded subsets of X.

According to Proposition 2 of Section 4, on every bounded set $B\in \mathcal{B}$, the order convergence structure (and, hence, ${\lambda }_{min}$) coincides with its bounded modification. That is, if $\left(x_{\alpha }\right)$ is a net contained in a bounded set B and

$$|x_{\alpha }-x|\stackrel{o}{\to }0,$$

(8)

then $\left(x_{\alpha }\right)$ converges to x in both ${\lambda }_{min}$ and $b{\lambda }_{min}$.

Since ${\lambda }_{min}$ is determined by order convergence over the entire space X and the bounded modification $b{\lambda }_{min}$ agrees with ${\lambda }_{min}$ on every bounded subset, as per (8), the modification does not introduce any additional convergent nets. Hence, we conclude that

$$b{\lambda }_{min}={\lambda }_{min}.$$

 □

The concept of a minimal locally solid convergence structure serves as a baseline for comparing various modified convergences. Any locally solid convergence structure that is not minimal necessarily refines order convergence, which can complicate properties such as duality and continuity.

Generalizations of the minimal convergence structure may be considered in broader contexts, including non-Archimedean vector lattices and spaces of operators. Such extensions have the potential to yield new insights into reflexivity, spectral theory, and ergodic behavior in vector lattices.

12. Fixed-Point Theory in Locally Solid Convergence Spaces

Definition 40 

(Fixed Point). Let $(X,\lambda )$ be a locally solid convergence space and let $T:X\to X$ be a mapping. A point $x\in X$ is called a fixed point of T if

$$T\left(x\right)=x.$$

Definition 41 

($\lambda$-Continuity). A mapping $T:X\to X$ is said to be $\lambda$-continuous if whenever a net $\left(x_{\alpha }\right)$ in X converges to x with respect to λ (written $x_{\alpha }\stackrel{\lambda }{\to }x$), then

$$T\left(x_{\alpha }\right)\stackrel{\lambda }{\to }T\left(x\right).$$

Definition 42 

(Contraction Mapping). Let $(X,\lambda )$ be a locally solid convergence space, where X is also a vector lattice. A mapping T: X$\to X$ is called a contraction mapping if there exists a constant $0 \le k<1$ such that for all $x,y\in X$,

$$|T(x)-T(y)| \le k|x-y|,$$

where the inequality is interpreted in the order structure of the vector lattice.

Definition 43 

(Complete Locally Solid Convergence Space). A locally solid convergence space $(X,\lambda )$ is called complete if every λ-Cauchy net in X converges with respect to λ.

Theorem 25 

(Banach Fixed-Point Theorem for Locally Solid Convergence Spaces). Let $(X,\lambda )$ be a complete Hausdorff locally solid convergence vector lattice and let T: X$\to X$ be a λ-continuous contraction mapping. Then, T has a unique fixed point in X. Moreover, for any initial point $x_0\in X$, the iterative sequence defined by

$$x_{n+1}=T\left(x_n\right),n\ge 0$$

converges to the fixed point with respect to λ.

Proof. 

Define the sequence $\left(x_n\right)$ by setting

$$x_{n+1}=T\left(x_n\right),n\ge 0,$$

with a given initial point $x_0\in X$. Since T is a contraction, there exists a constant k with $0 \le k<1$, such that for all $x,y\in X$,

$$|T(x)-T(y)| \le k|x-y|.$$

Applying this to $x=x_n$ and $y=x_{n-1}$, we obtain

$$|x_{n+1}-x_n| = |T\left(x_n\right)-T\left(x_{n-1}\right)| \le k|x_n-x_{n-1}|.$$

By induction, this yields

$$|x_{n+1}-x_n| \le k^n|x_1-x_0|\mathrm{for}\mathrm{all}n\ge 0.$$

Let $m>n$. Using the triangle inequality and the solidity of $\lambda$, we have

$$|x_m-x_n| \le \sum _{j=n}^{m-1}|x_{j+1}-x_j| \le |x_1-x_0|\sum _{j=n}^{m-1}{k}^j.$$

Since $0 \le k<1$, the geometric series ${\sum }_{j=0}^{\infty }{k}^j$ converges. Therefore, given any $\epsilon >0$, we can choose N large enough so that for all $m>n\ge N$,

$$|x_m-x_n| \le |x_1-x_0|\sum _{j=n}^{\infty }{k}^j<\epsilon .$$

Thus, $\left(x_n\right)$ is a $\lambda$-Cauchy sequence.

Since $(X,\lambda )$ is complete, there exists $x^{*}\in X$ such that

$$x_n\stackrel{\lambda }{\to }{x}^{*}.$$

The $\lambda$-continuity of T allows us to pass the limit through T, so that

$$T\left(x^{*}\right)=T\left(\underset{n\to \infty }{lim}{x}_n\right)=\underset{n\to \infty }{lim}T\left(x_n\right)=\underset{n\to \infty }{lim}{x}_{n+1}=x^{*}.$$

Hence, $x^{*}$ is a fixed point of T.

Suppose there exists another fixed point $y^{*}\in X$ such that $T\left(y^{*}\right)=y^{*}$. Then, by the contraction property,

$$|x^{*}-y^{*}| = |T\left(x^{*}\right)-T\left(y^{*}\right)| \le k|x^{*}-y^{*}|.$$

Since $0 \le k<1$, the only possibility is $|x^{*}-y^{*}|=0$, which implies that $x^{*}=y^{*}$. Thus, the fixed point is unique.

This completes the proof.  □

Definition 44 

(Uniformly Contractive Mapping). A mapping $T:X\to X$ is said to be uniformly contractive with respect to a family of regulators ${\left\{e_i\right\}}_{i\in I}\subseteq X_{+}$ if for each $i\in I$, there exists a constant $0 \le k_i<1$, such that for all $x,y\in X$,

$$|T\left(x\right)-T\left(y\right)| \wedge e_i \le k_i(|x-y| \wedge e_i).$$

Proposition 19. 

Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let $T:X\to X$ be uniformly contractive with respect to a family of regulators that generate the convergence structure λ. Then, T has a unique fixed point in X.

Proof. 

Since T is uniformly contractive, there exists a family of regulators ${\left\{e_i\right\}}_{i\in I}$ generating $\lambda$ and a constant $0 \le k<1$ such that for every $x,y\in X$ and each regulator $e_i$, we have

$$|T\left(x\right)-T\left(y\right)| \wedge e_i \le k\left(|x-y| \wedge e_i\right).$$

Let $x_0\in X$ be arbitrary and define the Picard iteration by

$$x_{n+1}=T\left(x_n\right),n\ge 0.$$

Then, for each $e_i$, by applying the contractivity, we obtain

$$|x_{n+1}-x_n| \wedge e_i=|T\left(x_n\right)-T\left(x_{n-1}\right)| \wedge e_i \le k\left(|x_n-x_{n-1}| \wedge e_i\right).$$

By induction, it follows that

$$|x_{n+1}-x_n| \wedge e_i \le k^n\left(|x_1-x_0| \wedge e_i\right).$$

Now, for any integers $m>n$, using the triangle inequality and the solidity of $\lambda$, we have

$$|x_m-x_n| \wedge e_i \le \sum _{j=n}^{m-1}|x_{j + 1}-x_j| \wedge e_i \le \left(|x_1-x_0| \wedge e_i\right)\sum _{j=n}^{m-1}{k}^j.$$

Since $0 \le k<1$, the series ${\sum }_{j=0}^{\infty }{k}^j$ converges. This implies that for each regulator $e_i$, the sequence $\left(x_n\right)$ is Cauchy with respect to the gauge induced by $e_i$. Hence, $\left(x_n\right)$ is a $\lambda$-Cauchy sequence.

By the completeness of $(X,\lambda )$, there exists an $x^{*}\in X$ such that

$$x_n\stackrel{\lambda }{\to }{x}^{*}.$$

Since T is $\lambda$-continuous, we have

$$T\left(x^{*}\right)=T\left(\underset{n\to \infty }{lim}{x}_n\right)=\underset{n\to \infty }{lim}T\left(x_n\right)=\underset{n\to \infty }{lim}{x}_{n+1}=x^{*},$$

so $x^{*}$ is a fixed point of T.

To prove uniqueness, assume that there exists another fixed point $y^{*}\in X$. Then, for each regulator $e_i$, we obtain

$$|x^{*}-y^{*}| \wedge e_i=|T\left(x^{*}\right)-T\left(y^{*}\right)| \wedge e_i \le k\left(|x^{*}-y^{*}| \wedge e_i\right).$$

Since $0 \le k<1$, it must be that

$$|x^{*}-y^{*}| \wedge e_i=0,$$

for every $e_i$. As the family ${\left\{e_i\right\}}_{i\in I}$ generates $\lambda$, it follows that $x^{*}=y^{*}$. Thus, the fixed point is unique.  □

The fixed-point results in locally solid convergence spaces extend classical fixed-point theory to settings where convergence is defined by order and lattice properties rather than by a norm or metric. This extension is particularly relevant in the study of nonlinear operators on vector lattices and has potential applications in differential and integral equations.

Future work could explore fixed-point theorems for mappings that are not globally contractive but that satisfy local contraction properties on appropriate bounded subsets or ideals. Additionally, the interplay between fixed-point theory and duality in locally solid convergence spaces may yield further insights into reflexivity and spectral theory.

13. Applications in Ergodic Theory

This section explores the implications of locally solid convergence structures in ergodic theory and the study of dynamical systems. By merging order-based convergence with dynamical systems theory, we obtain novel insights into the long-term behavior of operators on vector lattices.

Definition 45 

(Dynamical System). Let $(X,\lambda )$ be a locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous operator. The pair $(X,T)$ is called a dynamical system in the locally solid setting.

Definition 46 

(Ergodic Averages). For a given $x\in X$, the ergodic averages are defined by

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right),N\in \mathbb{N}.$$

Lemma 16 

(Ergodic Net is $\lambda$-Cauchy). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let T: X $\to X$ be a λ-continuous operator that is power-bounded (i.e., there exists $M>0$ such that $\parallel T^n\parallel \le M$ for all $n\in \mathbb{N}$). Then, for every $x\in X$, the sequence ${\left\{A_N\left(x\right)\right\}}_{N\in \mathbb{N}}$ of ergodic averages is λ-Cauchy.

Proof. 

For each $x\in X$, define the ergodic averages by

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

Let $P\in \mathbb{N}$ and consider the difference between two ergodic averages:

$$A_{N+P}\left(x\right)-A_N\left(x\right)={\displaystyle \frac{1}{N+P}}\sum _{n=0}^{N+P-1}{T}^n\left(x\right)-{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

We split the sum in the first term:

$$A_{N+P}\left(x\right)-A_N\left(x\right)=\left({\displaystyle \frac{1}{N + P}}-{\displaystyle \frac{1}{N}}\right)\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Observe that

$${\displaystyle \frac{1}{N+P}}-{\displaystyle \frac{1}{N}}=-{\displaystyle \frac{P}{N(N+P)}}.$$

Thus, we can rewrite the difference as

$$A_{N+P}\left(x\right)-A_N\left(x\right)=-{\displaystyle \frac{P}{N(N+P)}}\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Since T is power-bounded, we have for all $n\in \mathbb{N}$ that

$$\parallel T^n\left(x\right)\parallel \le M\parallel x\parallel ,$$

so that

$$∥\sum _{n=0}^{N-1}{T}^n\left(x\right)∥ \le NM\parallel x\parallel \mathrm{and}∥\sum _{n=N}^{N + P-1}{T}^n\left(x\right)∥ \le PM\parallel x\parallel .$$

Hence, by applying the triangle inequality, we obtain

$$\begin{array}{cc}\hfill ∥A_{N + P}\left(x\right)-A_N\left(x\right)∥& \le {\displaystyle \frac{P}{N(N + P)}}·∥\sum _{n=0}^{N-1}{T}^n\left(x\right)∥ + {\displaystyle \frac{1}{N + P}}·∥\sum _{n=N}^{N + P-1}{T}^n\left(x\right)∥\hfill \\ & \le {\displaystyle \frac{P}{N(N + P)}}·\left(NM\parallel x\parallel \right) + {\displaystyle \frac{PM\parallel x\parallel }{N + P}}\hfill \\ & ={\displaystyle \frac{PM\parallel x\parallel }{N + P}} + {\displaystyle \frac{PM\parallel x\parallel }{N + P}}\hfill \\ & ={\displaystyle \frac{2PM\parallel x\parallel }{N + P}}.\hfill \end{array}$$

Given any $\epsilon >0$, choose $N_0\in \mathbb{N}$ such that for all $N\ge N_0$ and all $P\ge 1$,

$${\displaystyle \frac{2PM\parallel x\parallel }{N+P}}<\epsilon .$$

Since the locally solid convergence $\lambda$ is generated by a family of regulators (or seminorms), this uniform estimate implies that for each regulator, the difference

$$|A_{N+P}\left(x\right)-A_N\left(x\right)|$$

can be made arbitrarily small in the $\lambda$-sense when N is sufficiently large. In other words, the net ${\left\{A_N\left(x\right)\right\}}_{N\in \mathbb{N}}$ is $\lambda$-Cauchy.

By the completeness of $(X,\lambda )$, there exists some $y\in X$ such that

$$A_N\left(x\right)\stackrel{\lambda }{\to }y.$$

Thus, the ergodic net is $\lambda$-Cauchy.  □

Theorem 26 

(Mean Ergodic Theorem for Locally Solid Convergence). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous, power-bounded operator. Then, for every $x\in X$, the ergodic averages

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)$$

converge in λ to some $P\left(x\right)\in X$. Moreover, the mapping $P:X\to Fix\left(T\right)$, defined by

$$P\left(x\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right),$$

is a λ-continuous projection onto the fixed-point subspace $Fix\left(T\right)=\{y\in X:T(y)=y\}$.

Proof. 

By Lemma 16, for each $x\in X$, the sequence ${\left\{A_N\left(x\right)\right\}}_{N\in \mathbb{N}}$ is $\lambda$-Cauchy. Since $(X,\lambda )$ is complete, there exists an element $P\left(x\right)\in X$ such that

$$A_N\left(x\right)\stackrel{\lambda }{\to }P\left(x\right)\mathrm{as}N\to \infty .$$

Thus, the mapping

$$P:X\to X,P\left(x\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)$$

is well defined.

Since T is $\lambda$-continuous and power-bounded, we have

$$T\left(P\left(x\right)\right)=T\left(\underset{N\to \infty }{lim}{A}_N\left(x\right)\right)=\underset{N\to \infty }{lim}T\left(A_N\left(x\right)\right).$$

Observe that

$$T\left(A_N\left(x\right)\right)=T\left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right).$$

By re-indexing the sum, we write

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N{T}^n\left(x\right)={\displaystyle \frac{1}{N}}\left(\sum _{n=0}^{N-1}{T}^n\left(x\right) + T^N\left(x\right)-x\right).$$

Thus,

$$T\left(A_N\left(x\right)\right)=A_N\left(x\right) + {\displaystyle \frac{T^N\left(x\right)-x}{N}}.$$

Since T is power-bounded (i.e., there exists $M>0$ such that $\parallel T^n\parallel \le M$ for all n), it follows that

$$∥{\displaystyle \frac{T^N\left(x\right)-x}{N}}∥ \le {\displaystyle \frac{M\parallel x\parallel + \parallel x\parallel }{N}}\to 0\mathrm{as}N\to \infty .$$

Therefore,

$$\underset{N\to \infty }{lim}T\left(A_N\left(x\right)\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)=P\left(x\right),$$

so that $T\left(P\right(x\left)\right)=P\left(x\right)$ and, hence, $P\left(x\right)\in Fix\left(T\right)$.

For any $x,y\in X$ and scalars $\alpha ,\beta$, by the linearity of T, we have

$$A_N(\alpha x + \beta y)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n(\alpha x + \beta y)=\alpha A_N\left(x\right) + \beta A_N\left(y\right).$$

Taking the $\lambda$-limit as $N\to \infty$, it follows that

$$P(\alpha x + \beta y)=\alpha P\left(x\right) + \beta P\left(y\right),$$

so that P is linear.

Next, note that if $x\in Fix\left(T\right)$, then $T^n\left(x\right)=x$ for all $n\ge 0$; hence,

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}x=x,$$

which implies that $P\left(x\right)=x$. Moreover, for an arbitrary $x\in X$, since $P\left(x\right)\in Fix\left(T\right)$, we have

$$P\left(P\right(x\left)\right)=P\left(x\right),$$

showing that P is idempotent. Thus, P is a projection onto $Fix\left(T\right)$.

The $\lambda$-continuity of P follows from the uniform estimates on the ergodic averages on bounded sets. Since T is $\lambda$-continuous and power-bounded, the convergence

$$A_N\left(x\right)\stackrel{\lambda }{\to }P\left(x\right)$$

is uniform on bounded sets (with respect to the family of regulators generating $\lambda$). Hence, P is $\lambda$-continuous.

Combining the above steps, we conclude that for every $x\in X$, the ergodic averages converge in $\lambda$ to $P\left(x\right)$, and the mapping

$$P:X\to Fix\left(T\right)$$

is a $\lambda$-continuous projection. This completes the proof.  □

Proposition 20 

(Uniform Ergodicity). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous, power-bounded operator. Assume that the ergodic averages $A_N\left(x\right)$ converge uniformly on every bounded subset of X. Then, the projection P defined in Theorem 26 is uniformly continuous on bounded sets.

Proof. 

Let $B\subset X$ be any bounded subset and let $\epsilon >0$. By the uniform convergence assumption, there exists an $N_0\in \mathbb{N}$, such that for all $N\ge N_0$, we have

$$\underset{x\in B}{sup}\parallel A_N\left(x\right)-P\left(x\right)\parallel <{\displaystyle \frac{\epsilon }{3}}.$$

Fix such an $N\ge N_0$. Recall that

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

Since T is $\lambda$-continuous and power-bounded, each iterate $T^n$ is $\lambda$-continuous. Hence, the finite sum defining $A_N\left(x\right)$ is uniformly continuous on B. In particular, there exists a $\delta >0$ such that for all $x,y\in B$ with $\parallel x-y\parallel <\delta$, we have

$$\parallel A_N\left(x\right)-A_N\left(y\right)\parallel <{\displaystyle \frac{\epsilon }{3}}.$$

Now, for any $x,y\in B$ with $\parallel x-y\parallel <\delta$, we have

$$\begin{array}{cc}\hfill \parallel P\left(x\right)-P\left(y\right)\parallel & \le \parallel P\left(x\right)-A_N\left(x\right)\parallel + \parallel A_N\left(x\right)-A_N\left(y\right)\parallel + \parallel A_N\left(y\right)-P\left(y\right)\parallel \hfill \\ & <{\displaystyle \frac{\epsilon }{3}} + {\displaystyle \frac{\epsilon }{3}} + {\displaystyle \frac{\epsilon }{3}}=\epsilon .\hfill \end{array}$$

This shows that P is uniformly continuous on the bounded set B. Since B was arbitrary, P is uniformly continuous on every bounded subset of X.  □

Definition 47 

(Nonlinear Ergodic Averages). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous nonlinear operator. For any $x\in X$, define the nonlinear ergodic averages by

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right),N\in \mathbb{N}.$$

Lemma 17 

(Nonlinear Cauchy Property). Assume that T is power-bounded and that there exists a constant $0 \le k<1$ such that for all $x,y\in X$ and all regulators $u\in X_{+}$,

$$|T\left(x\right)-T\left(y\right)| \wedge u \le k\left(|x-y| \wedge u\right).$$

Then, for every $x\in X$, the sequence ${\left\{A_N\left(x\right)\right\}}_{N\in \mathbb{N}}$ is λ-Cauchy.

Proof. 

Let $x\in X$ be arbitrary. By definition, for each $N\in \mathbb{N}$, we set

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

To show that $\left\{A_N\left(x\right)\right\}$ is $\lambda$-Cauchy, we must prove that for every regulator $u\in X^{+}$ and every $\epsilon >0$, there exists an $N_0\in \mathbb{N}$ such that for all $N\ge N_0$ and every $P\ge 1$,

$$\left|A_{N + P}\left(x\right)-A_N\left(x\right)\right| \wedge u<\epsilon .$$

Write the difference between two ergodic averages:

$$A_{N+P}\left(x\right)-A_N\left(x\right)={\displaystyle \frac{1}{N+P}}\sum _{n=0}^{N+P-1}{T}^n\left(x\right)-{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

We split this difference into two parts:

$$A_{N+P}\left(x\right)-A_N\left(x\right)=\left({\displaystyle \frac{1}{N + P}}-{\displaystyle \frac{1}{N}}\right)\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Note that

$${\displaystyle \frac{1}{N+P}}-{\displaystyle \frac{1}{N}}=-{\displaystyle \frac{P}{N(N+P)}}.$$

Hence, the absolute value (measured with the regulator u) of the first term is bounded by

$$\left|\left({\displaystyle \frac{1}{N + P}}-{\displaystyle \frac{1}{N}}\right)\sum _{n=0}^{N-1}{T}^n\left(x\right)\right| \wedge u \le {\displaystyle \frac{P}{N(N+P)}}\sum _{n=0}^{N-1}\left|T^n\left(x\right)\right| \wedge u.$$

Because T is power-bounded, there exists a constant $M>0$ (depending on x and the regulator u), such that for all n,

$$\left|T^n\left(x\right)\right| \wedge u \le M.$$

Thus, the first term is bounded by

$${\displaystyle \frac{P}{N(N+P)}}·NM={\displaystyle \frac{PM}{N+P}}.$$

For the second term, we have

$${\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}\left|T^n\left(x\right)\right| \wedge u \le {\displaystyle \frac{PM}{N+P}}.$$

Combining these two estimates yields

$$\left|A_{N + P}\left(x\right)-A_N\left(x\right)\right| \wedge u \le {\displaystyle \frac{PM}{N+P}} + {\displaystyle \frac{PM}{N+P}}={\displaystyle \frac{2PM}{N+P}}.$$

Given any $\epsilon >0$, one can choose an N sufficiently large so that for all $P\ge 1$,

$${\displaystyle \frac{2PM}{N+P}}<\epsilon .$$

This shows that the difference $\left|A_{N + P}\left(x\right)-A_N\left(x\right)\right| \wedge u$ can be made arbitrarily small, which means that $\left\{A_N\left(x\right)\right\}$ is a $\lambda$-Cauchy sequence.  □

Theorem 27 

(Mean Ergodic Theorem for Nonlinear Operators). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and let $T:X\to X$ be a λ-continuous nonlinear operator satisfying the contraction condition above. Then, for every $x\in X$, the ergodic averages

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)$$

converge in λ to a limit $P\left(x\right)\in X$, and the mapping $P:X\to Fix\left(T\right)$ defined by

$$P\left(x\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)$$

is a λ-continuous projection onto the fixed-point set $Fix\left(T\right)=\{y\in X:T(y)=y\}$.

Proof. 

For any $x\in X$, define the ergodic averages by

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

We show that the net $\left\{A_N\left(x\right)\right\}$ is $\lambda$-Cauchy. For any $P\in \mathbb{N}$, consider the difference

$$A_{N+P}\left(x\right)-A_N\left(x\right)={\displaystyle \frac{1}{N+P}}\sum _{n=0}^{N+P-1}{T}^n\left(x\right)-{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

Split the first sum into two parts:

$$A_{N+P}\left(x\right)-A_N\left(x\right)=\left({\displaystyle \frac{1}{N + P}}-{\displaystyle \frac{1}{N}}\right)\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Since

$${\displaystyle \frac{1}{N+P}}-{\displaystyle \frac{1}{N}}=-{\displaystyle \frac{P}{N(N+P)}},$$

we have

$$A_{N+P}\left(x\right)-A_N\left(x\right)=-{\displaystyle \frac{P}{N(N+P)}}\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Under the contraction condition and the $\lambda$-continuity of T, the sequence $\left\{T^n\left(x\right)\right\}$ is uniformly bounded (in the sense of each regulator). Hence, there exists a constant $C>0$ (depending on x and the chosen regulator), such that

$${∥\sum _{n=0}^{N-1}{T}^n\left(x\right)∥}_e \le NC\mathrm{and}{∥\sum _{n=N}^{N + P-1}{T}^n\left(x\right)∥}_e \le PC,$$

where ${\parallel ·\parallel }_e$ denotes the seminorm associated with a regulator e. Therefore,

$$\parallel A_{N+P}\left(x\right)-A_N{\left(x\right)\parallel }_e \le {\displaystyle \frac{P}{N(N+P)}}\left(NC\right) + {\displaystyle \frac{1}{N+P}}\left(PC\right)={\displaystyle \frac{2PC}{N+P}}.$$

Given any $\epsilon >0$, one can choose an N sufficiently large so that for every $P\ge 1$,

$${\displaystyle \frac{2PC}{N+P}}<\epsilon .$$

(9)

Thus, $\left\{A_N\left(x\right)\right\}$ is $\lambda$-Cauchy.

Since $(X,\lambda )$ is complete, every $\lambda$-Cauchy net converges. Therefore, there exists an element

$$P\left(x\right)\in X$$

such that

$$A_N\left(x\right)\stackrel{\lambda }{\to }P\left(x\right)\mathrm{as}N\to \infty .$$

We can now show that $P\left(x\right)$ is a fixed point of T. By the $\lambda$-continuity of T, we have

$$T\left(P\left(x\right)\right)=T\left(\underset{N\to \infty }{lim}{A}_N\left(x\right)\right)=\underset{N\to \infty }{lim}T\left(A_N\left(x\right)\right).$$

Note that

$$T\left(A_N\left(x\right)\right)=T\left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}T\left(T^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right).$$

Re-indexing the sum yields

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N{T}^n\left(x\right)=A_N\left(x\right) + {\displaystyle \frac{T^N\left(x\right)-x}{N}}.$$

Under the contraction condition, the term ${\displaystyle \frac{T^N\left(x\right)-x}{N}}$ converges to 0 in $\lambda$ as $N\to \infty$. Hence,

$$T\left(P\left(x\right)\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)=P\left(x\right),$$

so that $P\left(x\right)\in Fix\left(T\right)$.

If $x\in Fix\left(T\right)$, then $T\left(x\right)=x$, so that $T^n\left(x\right)=x$ for all $n\ge 0$ and

$$A_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}x=x.$$

Thus, $P\left(x\right)=x$, showing that P acts as the identity on $Fix\left(T\right)$ and, hence, P is a projection. Moreover, the uniform estimates used in (9) (which hold uniformly on bounded sets) imply that the mapping $P:X\to Fix\left(T\right)$ is $\lambda$-continuous.

For every $x\in X$, the ergodic averages $A_N\left(x\right)$ converge in $\lambda$ to a limit $P\left(x\right)$, which is a fixed point of T. The mapping

$$P:X\to Fix\left(T\right),P\left(x\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)$$

is a $\lambda$-continuous projection onto the fixed-point set $Fix\left(T\right)=\{y\in X:T(y)=y\}$. This completes the proof.  □

The above result extends classical mean ergodic theory to the nonlinear setting, suggesting new fixed point and stability phenomena in nonlinear dynamics.

Definition 48 

(p-adic Dynamical System). Let $(X,{\lambda }_p)$ be a locally solid convergence vector lattice over the field of p-adic numbers ${\mathbb{Q}}_p$, equipped with the p-adic norm ${|·|}_p$. A mapping $T:X\to X$ is called a p-adic dynamical system if it is ${\lambda }_p$-continuous and power-bounded with respect to the ultrametric induced by ${|·|}_p$.

Definition 49 

(p-adic Ergodic Averages). For $x\in X$, define the p-adic ergodic averages as

$$A_N^p\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right),N\in \mathbb{N}.$$

Theorem 28 

(p-adic Mean Ergodic Theorem). Let $(X,{\lambda }_p)$ be a complete p-adic locally solid convergence vector lattice and let $T:X\to X$ be a ${\lambda }_p$-continuous, power-bounded operator. Then, for every $x\in X$, the ergodic averages $A_N^p\left(x\right)$ converge in ${\lambda }_p$ to a limit $P_p\left(x\right)$ and the projection $P_p:X\to Fix\left(T\right)$ is ${\lambda }_p$-continuous.

Proof. 

For any fixed $x\in X$, define the p-adic ergodic averages by

$$A_N^p\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

For integers N and $P\ge 1$, consider the difference

$$A_{N+P}^p\left(x\right)-A_N^p\left(x\right)={\displaystyle \frac{1}{N+P}}\sum _{n=0}^{N+P-1}{T}^n\left(x\right)-{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right).$$

Splitting the first sum yields

$$A_{N+P}^p\left(x\right)-A_N^p\left(x\right)=\left({\displaystyle \frac{1}{N + P}}-{\displaystyle \frac{1}{N}}\right)\sum _{n=0}^{N-1}{T}^n\left(x\right) + {\displaystyle \frac{1}{N+P}}\sum _{n=N}^{N+P-1}{T}^n\left(x\right).$$

Noting that

$${\displaystyle \frac{1}{N+P}}-{\displaystyle \frac{1}{N}}=-{\displaystyle \frac{P}{N(N+P)}}$$

and using the ultrametric (non-Archimedean) property of the p-adic norm, we obtain

$$\parallel A_{N+P}^p\left(x\right)-A_N^p{\left(x\right)\parallel }_p \le max\left\{{\displaystyle \frac{P}{{|N(N + P)|}_p}}\underset{0 \le n<N}{max}\parallel T^n{\left(x\right)\parallel }_p,{\displaystyle \frac{1}{{|N + P|}_p}}\underset{N \le n<N + P}{max}{\parallel T^n\left(x\right)\parallel }_p\right\}.$$

Since T is power-bounded, there exists a constant $M>0$ such that

$$\parallel T^n{\left(x\right)\parallel }_p \le M{\parallel x\parallel }_p\mathrm{for}\mathrm{all}n\ge 0.$$

Thus, both terms in the maximum are bounded by quantities that tend to zero as $N\to \infty$. In particular, for every $\epsilon >0$, one may choose an N sufficiently large so that for every $P\ge 1$,

$$\parallel A_{N+P}^p\left(x\right)-A_N^p{\left(x\right)\parallel }_p<\epsilon .$$

(10)

Hence, the net ${\left\{A_N^p\left(x\right)\right\}}_{N\in \mathbb{N}}$ is ${\lambda }_p$-Cauchy.

By the completeness of $(X,{\lambda }_p)$, every ${\lambda }_p$-Cauchy net-converges. Therefore, for each $x\in X$, there exists

$$P_p\left(x\right)\in X\mathrm{such}\mathrm{that}{A}_N^p\left(x\right)\stackrel{{\lambda }_p}{\to }{P}_p\left(x\right)\mathrm{as}N\to \infty .$$

This defines the mapping

$$P_p:X\to X,P_p\left(x\right)=\underset{N\to \infty }{lim}{A}_N^p\left(x\right).$$

We can now show that $P_p\left(x\right)$ is a fixed point of T. By the ${\lambda }_p$-continuity of T,

$$T\left(P_p\left(x\right)\right)=T\left(\underset{N\to \infty }{lim}{A}_N^p\left(x\right)\right)=\underset{N\to \infty }{lim}T\left(A_N^p\left(x\right)\right).$$

Observe that

$$T\left(A_N^p\left(x\right)\right)=T\left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right).$$

Re-indexing the sum gives

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^{n+1}\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N{T}^n\left(x\right)=A_N^p\left(x\right) + {\displaystyle \frac{T^N\left(x\right)-x}{N}}.$$

Since T is power-bounded, the term ${\displaystyle \frac{T^N\left(x\right)-x}{N}}$ converges to 0 in the p-adic sense as $N\to \infty$. Therefore,

$$T\left(P_p\left(x\right)\right)=\underset{N\to \infty }{lim}{A}_N^p\left(x\right)=P_p\left(x\right),$$

so that $P_p\left(x\right)$ belongs to the fixed-point set

$$Fix\left(T\right)=\{y\in X:T(y)=y\}.$$

Finally, we can show that the mapping $P_p:X\to Fix\left(T\right)$ is ${\lambda }_p$-continuous. The uniform estimates in (10) (which hold uniformly on bounded sets in the p-adic topology) imply that if x and y are close in the ${\lambda }_p$-sense, then their ergodic averages $A_N^p\left(x\right)$ and $A_N^p\left(y\right)$ are uniformly close (for large N). Passing to the limit as $N\to \infty$, it follows that

$$\parallel P_p\left(x\right)-P_p{\left(y\right)\parallel }_p$$

can be made arbitrarily small when ${\parallel x-y\parallel }_p$ is small. Hence, $P_p$ is ${\lambda }_p$-continuous.

For every $x\in X$, the ergodic averages $A_N^p\left(x\right)$ converge in ${\lambda }_p$ to a limit $P_p\left(x\right)$ and the mapping

$$P_p:X\to Fix\left(T\right),P_p\left(x\right)=\underset{N\to \infty }{lim}{A}_N^p\left(x\right)$$

is a ${\lambda }_p$-continuous projection onto $Fix\left(T\right)$. This completes the proof.  □

The ultrametric structure of ${\mathbb{Q}}_p$ often enhances convergence properties. In this context, the ergodic averages stabilize more rapidly, providing deeper insights into invariant measures and ergodic behavior in the p-adic framework.

Definition 50 

(Dual Ergodic Projection). Let $(X,\lambda )$ be a locally solid convergence vector lattice with dual space $X^{*}$ (equipped with the topology of uniform convergence on bounded sets). For a λ-continuous operator $T:X\to X$, the dual ergodic projection is the operator

$$P^{*}:X^{*}\to X^{*},P^{*}\left(f\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{\left(T^{*}\right)}^n\left(f\right),$$

where $T^{*}:X^{*}\to X^{*}$ is the dual operator.

Lemma 18 

(Convergence in the Dual Space). If $T:X\to X$ is a λ-continuous, power-bounded operator and $P:X\to Fix\left(T\right)$ is the ergodic projection, then for every $f\in X^{*}$, the sequence ${\left\{{\left(T^{*}\right)}^n\left(f\right)\right\}}_{n\in \mathbb{N}}$ converges in the dual convergence structure to a unique limit in $X^{*}$.

Proof. 

Since T is power-bounded, there exists a constant $M>0$ such that for all $n\in \mathbb{N}$,

$$\parallel T^n\parallel \le M.$$

Thus, for any $f\in X^{*}$ and all $n\in \mathbb{N}$,

$$\parallel {\left(T^{*}\right)}^n\left(f\right)\parallel =\underset{\parallel x\parallel \le 1}{sup}|{\left(T^{*}\right)}^n\left(f\right)\left(x\right)| =\underset{\parallel x\parallel \le 1}{sup}|f\left(T^n\left(x\right)\right)| \le \parallel f\parallel ·\parallel T^n\parallel \le M\parallel f\parallel .$$

Hence, the sequence $\left\{{\left(T^{*}\right)}^n\left(f\right)\right\}$ is uniformly bounded in $X^{*}$.

Recall that convergence in $X^{*}$ (with respect to the dual convergence structure) is given by uniform convergence on each bounded subset of X. Let $\left\{B_{\alpha }\right\}$ be a family of bounded sets that generate the topology on $X^{*}$. We claim that for each bounded set $B_{\alpha }$ and every $\epsilon >0$, there exists an $N\in \mathbb{N}$ such that for all $m,n\ge N$,

$$\underset{x\in B_{\alpha }}{sup}\left|{\left(T^{*}\right)}^n\left(f\right)\left(x\right)-{\left(T^{*}\right)}^m\left(f\right)\left(x\right)\right|<\epsilon .$$

Since ${\left(T^{*}\right)}^n\left(f\right)\left(x\right)=f\left(T^n\left(x\right)\right)$, the difference

$$\left|{\left(T^{*}\right)}^n\left(f\right)\left(x\right)-{\left(T^{*}\right)}^m\left(f\right)\left(x\right)\right|=\left|f\left(T^n\left(x\right)\right)-f\left(T^m\left(x\right)\right)\right|$$

can be controlled by the uniform continuity of f on the bounded set $B_{\alpha }$, provided that the iterates $\left\{T^n\left(x\right)\right\}$ do not oscillate too much for $x\in B_{\alpha }$. The existence of the ergodic projection P, together with the power-boundedness and $\lambda$-continuity of T, ensures that the oscillations of $T^n\left(x\right)$ are eventually damped in a uniform manner. Consequently, the sequence $\left\{{\left(T^{*}\right)}^n\left(f\right)\right\}$ is Cauchy in the dual convergence structure.

Since the dual space $X^{*}$ is complete with respect to the dual convergence structure, every Cauchy sequence converges. Thus, there exists a unique $g\in X^{*}$ such that

$${\left(T^{*}\right)}^n\left(f\right)\stackrel{{\lambda }^{*}}{\to }g\mathrm{as}n\to \infty .$$

Uniqueness follows from the Hausdorff property of the dual convergence: if two limits existed, their difference would vanish on every bounded set, forcing them to be equal.

We conclude that for every $f\in X^{*}$, the sequence $\left\{{\left(T^{*}\right)}^n\left(f\right)\right\}$ converges in the dual convergence structure to a unique limit in $X^{*}$. This completes the proof.  □

Theorem 29 

(Ergodic Duality Theorem). Let $(X,\lambda )$ be a complete locally solid convergence vector lattice and $T:X\to X$ be a λ-continuous, power-bounded operator. Then, the ergodic projection $P:X\to Fix\left(T\right)$ satisfies

$$f\left(P\left(x\right)\right)=P^{*}\left(f\right)\left(x\right)\mathit{for}\mathit{all}x\in X\mathit{and}f\in X^{*}.$$

In particular, the duality between X and $X^{*}$ is preserved under ergodic averaging.

Proof. 

By definition, the ergodic projection P on X is given by

$$P\left(x\right)=\underset{N\to \infty }{lim}{A}_N\left(x\right)\mathrm{with}{A}_N\left(x\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right),$$

while the dual ergodic projection $P^{*}$ on $X^{*}$ is defined as

$$P^{*}\left(f\right)=\underset{N\to \infty }{lim}{A}_N^{*}\left(f\right)\mathrm{with}{A}_N^{*}\left(f\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{\left(T^{*}\right)}^n\left(f\right),$$

where $T^{*}:X^{*}\to X^{*}$ is the dual operator defined by $\left(T^{*}f\right)\left(x\right)=f\left(T\left(x\right)\right)$.

For any $x\in X$ and $f\in X^{*}$, the continuity of f (with respect to $\lambda$) yields

$$f\left(P\left(x\right)\right)=f\left(\underset{N\to \infty }{lim}{A}_N\left(x\right)\right)=\underset{N\to \infty }{lim}f\left(A_N\left(x\right)\right).$$

Using the linearity of f, we have

$$f\left(A_N\left(x\right)\right)=f\left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{T}^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right).$$

Since, by definition,

$$f\left(T^n\left(x\right)\right)={\left(T^{*}\right)}^n\left(f\right)\left(x\right),$$

we can write

$${\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f\left(T^n\left(x\right)\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{\left(T^{*}\right)}^n\left(f\right)\left(x\right)=A_N^{*}\left(f\right)\left(x\right).$$

Taking the limit as $N\to \infty$, it follows that

$$f\left(P\left(x\right)\right)=\underset{N\to \infty }{lim}{A}_N^{*}\left(f\right)\left(x\right)=P^{*}\left(f\right)\left(x\right).$$

Thus, for every $x\in X$ and every $f\in X^{*}$, we can show that

$$f\left(P\left(x\right)\right)=P^{*}\left(f\right)\left(x\right),$$

which implies that the duality between X and $X^{*}$ is preserved under ergodic averaging.  □

This result links the ergodic behavior in the primal space to that in the dual space, offering new perspectives on spectral decompositions and long-term behavior in ordered Banach spaces.

Definition 51 

(Ergodic Solution of a Differential Equation). Consider a differential equation of the form

$${\displaystyle \frac{du}{dt}}=F\left(u\right),u\left(0\right)=u_0,$$

where F is a nonlinear operator acting on a function space X endowed with a locally solid convergence structure λ. A function $u^{*}\in X$ is called an ergodic solution if there exists a time-averaging process

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{T}}{\int }_0^{T}u\left(t\right)dt=u^{*},$$

where $u\left(t\right)$ is the solution of the differential equation with initial condition $u_0$ and the limit is taken in the λ-sense.

Proposition 21 

(Existence of Invariant Solutions). Let X be a function space with a locally solid convergence structure λ and let $F:X\to X$ be a λ-continuous operator such that the associated semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$, defined by $S\left(t\right)u_0=u\left(t\right)$, is power-bounded. Then, the ergodic averages

$$A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)u_{0}dt$$

converge in λ to an invariant function $u^{*}\in X$, which is an ergodic solution of the differential equation.

Proof. 

Since the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is power-bounded, there exists a constant $M>0$ such that for every $t\ge 0$ and every regulator e generating $\lambda$,

$$\parallel S\left(t\right)u_0{\parallel }_e \le M{\parallel u_0\parallel }_e.$$

Thus, for each $T>0$,

$$\parallel A_T\left(u_0\right){\parallel }_e={∥{\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)u_{0}dt∥}_e \le {\displaystyle \frac{1}{T}}{\int }_0^T\parallel S\left(t\right)u_0{\parallel }_{e}dt \le M{\parallel u_0\parallel }_e.$$

This shows that the net ${\left\{A_T\left(u_0\right)\right\}}_{T>0}$ is uniformly bounded.

Next, let $\epsilon >0$. By the $\lambda$-continuity of the semigroup and standard estimates (which rely on power-boundedness), one may show that there exists a $T_0>0$ such that for all $T,T^{\prime }\ge T_0$,

$$\parallel A_T\left(u_0\right)-A_{T^{\prime }}\left(u_0\right){\parallel }_e<\epsilon$$

for every regulator e. In other words, the net $\left\{A_T\left(u_0\right)\right\}$ is $\lambda$-Cauchy.

Since X is complete with respect to the locally solid convergence structure $\lambda$, every $\lambda$-Cauchy net converges. Hence, there exists a limit

$$u^{*}\in X\mathrm{such}\mathrm{that}{A}_T\left(u_0\right)\stackrel{\lambda }{\to }{u}^{*}\mathrm{as}T\to \infty .$$

We now show that the limit $u^{*}$ is invariant under the semigroup. For any fixed $t\ge 0$, consider

$$S\left(t\right)A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)S\left(s\right)u_{0}ds={\displaystyle \frac{1}{T}}{\int }_0^{T}S(t + s)u_{0}ds.$$

Changing the variable via $r=t + s$, we can write

$$S\left(t\right)A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_t^{T + t}S\left(r\right)u_{0}dr.$$

This expression may be split as

$${\displaystyle \frac{1}{T}}{\int }_t^{T + t}S\left(r\right)u_{0}dr={\displaystyle \frac{1}{T}}{\int }_0^{T + t}S\left(r\right)u_{0}dr-{\displaystyle \frac{1}{T}}{\int }_0^{t}S\left(r\right)u_{0}dr.$$

Since

$$A_{T + t}\left(u_0\right)={\displaystyle \frac{1}{T + t}}{\int }_0^{T + t}S\left(r\right)u_{0}dr\mathrm{and}{\displaystyle \frac{1}{T}}{\int }_0^{t}S\left(r\right)u_{0}dr\to 0\mathrm{as}T\to \infty ,$$

it follows that

$$S\left(t\right)A_T\left(u_0\right)\stackrel{\lambda }{\to }{u}^{*}\mathrm{as}T\to \infty .$$

Using the $\lambda$-continuity of $S\left(t\right)$ and the uniqueness of limits, we deduce that

$$S\left(t\right)u^{*}=u^{*}\mathrm{for}\mathrm{all}t\ge 0.$$

Thus, $u^{*}$ is invariant, i.e., $u^{*}\in Fix\left(S\left(t\right)\right)$.

The invariance $S\left(t\right)u^{*}=u^{*}$ means that $u^{*}$ is an equilibrium (or ergodic) solution of the differential equation associated with the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$.

We see that the ergodic averages $A_T\left(u_0\right)$ converge in $\lambda$ to an invariant function $u^{*}\in X$, which is an ergodic solution of the differential equation. This completes the proof.  □

Theorem 30 

(Ergodic Differential Equation Theorem). Let X be a Banach lattice with a locally solid convergence structure λ and let $F:X\to X$ be a λ-continuous operator generating a semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ that is power-bounded. Then, for every initial condition $u_0\in X$, the time averages

$$A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)u_{0}dt$$

converge in λ to a unique ergodic solution $u^{*}$ of the differential equation ${\displaystyle \frac{du}{dt}}=F\left(u\right)$.

Proof. 

Since ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is power-bounded, there exists a constant $M>0$ such that

$$\parallel S\left(t\right)u_0\parallel \le M\parallel u_0\parallel \mathrm{for}\mathrm{all}t\ge 0.$$

Thus, for every $T>0$, we have

$$\parallel A_T\left(u_0\right)\parallel =∥{\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)u_{0}dt∥ \le {\displaystyle \frac{1}{T}}{\int }_0^T\parallel S\left(t\right)u_0\parallel dt \le M\parallel u_0\parallel .$$

This uniform bound (with respect to the seminorms generating $\lambda$) implies that the net ${\left\{A_T\left(u_0\right)\right\}}_{T>0}$ is bounded. Moreover, using the $\lambda$-continuity of the semigroup and standard estimates, one can show that for every $\epsilon >0$, there exists $T_0>0$, such that for all $T,T^{\prime }\ge T_0$,

$$\parallel A_T\left(u_0\right)-A_{T^{\prime }}\left(u_0\right){\parallel }_{\lambda }<\epsilon .$$

In other words, the net $\left\{A_T\left(u_0\right)\right\}$ is $\lambda$-Cauchy.

Since X is complete with respect to the locally solid convergence structure $\lambda$, every $\lambda$-Cauchy net converges. Therefore, there exists an element

$$u^{*}\in X$$

such that

$$A_T\left(u_0\right)\stackrel{\lambda }{\to }{u}^{*}\mathrm{as}T\to \infty .$$

We can now show that the limit $u^{*}$ is invariant under the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$. For any fixed $t\ge 0$, consider

$$S\left(t\right)A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)S\left(s\right)u_{0}ds={\displaystyle \frac{1}{T}}{\int }_0^{T}S(t + s)u_{0}ds.$$

Changing the variable by setting $r=t + s$, we obtain

$$S\left(t\right)A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_t^{T + t}S\left(r\right)u_{0}dr.$$

Using the power-boundedness and $\lambda$-continuity of the semigroup, one can show that the difference

$$\parallel S\left(t\right)A_T\left(u_0\right)-A_T\left(u_0\right){\parallel }_{\lambda }$$

tends to zero as $T\to \infty$. Passing to the limit and using the $\lambda$-continuity of $S\left(t\right)$, we deduce that

$$S\left(t\right)u^{*}=u^{*}\mathrm{for}\mathrm{all}t\ge 0.$$

Thus, $u^{*}$ is invariant and belongs to the fixed-point set of the semigroup (or equivalently, is an equilibrium of the differential equation).

The invariance $S\left(t\right)u^{*}=u^{*}$ implies that $u^{*}$ is a stationary (or ergodic) solution of the differential equation

$${\displaystyle \frac{du}{dt}}=F\left(u\right)$$

since the semigroup $\left\{S\right(t\left)\right\}$ is generated by F. Moreover, by the Hausdorff property of the $\lambda$-topology, the limit $u^{*}$ is unique. Hence, the time averages $A_T\left(u_0\right)$ converge in $\lambda$ to the unique ergodic solution $u^{*}$.

Combining these steps, we conclude that for every $u_0\in X$, the time averages

$$A_T\left(u_0\right)={\displaystyle \frac{1}{T}}{\int }_0^{T}S\left(t\right)u_{0}dt$$

converge in $\lambda$ to a unique ergodic solution $u^{*}$ of the differential equation ${\displaystyle \frac{du}{dt}}=F\left(u\right)$.  □

Time averaging in differential equations provides a powerful method for studying the long-term behavior of solutions. In the locally solid convergence setting, this method not only guarantees the existence of invariant solutions but also links the dynamical properties of the semigroup with the underlying order structure of the space.

14. Conclusions

In this work, we significantly broadened the classical framework of locally solid convergence structures in vector lattices. Beginning with the traditional theory of locally solid topologies [1], we introduced a range of modifications—most notably the unbounded and bornological modifications—that extend the scope of convergence beyond the confines of standard topological approaches. These extensions not only preserve the essential order-based characteristics inherent in vector lattices but also provide a unified framework to handle both topological and non-topological convergence phenomena.

Our investigation has demonstrated that the unbounded modification captures convergence behavior through regulators from the positive cone, while the bornological approach localizes convergence to bounded subsets. These results have far-reaching implications in functional analysis, as evidenced by our applications to operator-induced convergence, duality, and fixed point theory [20,21]. In particular, the operator–theoretic aspects reveal that positive operators preserve convergence in this enriched setting, thereby extending classical results to a more general context.

Furthermore, we extended these ideas to non-Archimedean settings, including ultrametric and p-adic frameworks, thereby opening new avenues for research in p-adic functional analysis and ergodic theory [13]. Our study of the topological modifications of non-topological convergence structures shows that under suitable conditions, local solidity is preserved, a result that bridges the gap between order convergence and its topological analogs.

The introduction of minimal locally solid convergence structures provides a canonical baseline, allowing for the systematic comparison of various convergence modifications. These generalizations, including Choquet modifications, not only enrich the theoretical landscape but also pave the way for novel applications in nonlinear analysis and operator theory.

Future research directions include a deeper exploration of duality and spectral theory in both Archimedean and non-Archimedean contexts, as well as the development of advanced fixed-point theorems for nonlinear operators within these generalized frameworks. We believe that the extended framework presented in this paper will serve as a robust foundation for ongoing and future investigations into the convergence theory of vector lattices.