On Topological Structures and Mapping Theorems in Intuitionistic Fuzzy 2-Normed Spaces

Abstract

In intuitionistic fuzzy 2-normed spaces, there are numerous symmetries in the topological structures and mapping theorems. In this work, we present the concept of an intuitionistic fuzzy 2-normed space(IF2NS) and demonstrate its structural properties using illustrative examples. This approach unifies and broadens the scope of both classical 2-normed spaces and intuitionistic fuzzy normed spaces when specific conditions are met. We introduce the idea of fuzzy open balls and explore the convergence of sequences with respect to the topology derived from the intuitionistic fuzzy 2-norm. In addition, we define left and right N-Cauchy sequences relative to the topologies ${\tau }_N$ and ${\tau }_{N^{-1}}$ and analyze their convergence characteristics. Special attention is given to the inherent symmetry of the 2-norm, where the magnitude of a pair of vectors remains invariant under exchange of arguments, and to the balanced interaction between membership and non-membership functions in the intuitionistic fuzzy setting. This intrinsic symmetry is further reflected in the proofs of the open mapping and closed graph theorems, which naturally preserve the symmetric structure of the underlying space The paper culminates with the formulation and proof of the open mapping theorem that can be considered for its symmetric properties and the closed graph theorem in the context of IF2NS, thereby generalizing essential theorems of functional analysis to this fuzzy setting.

1. Introduction

In 1965, Zadeh proposed the foundational theory of fuzzy sets, initiating a new paradigm for handling uncertainty in mathematical structures [1]. This idea prompted the formulation of fuzzy analogues of classical analytical structures. One such extension was the fuzzy norm for linear spaces, first explored by Katsaras and later refined by Felbin [2], who presented an alternate approach incorporating generalized metrics of the Kaleva–Seikkala type. Cheng and Mordeson subsequently introduced fuzzy linear operators and extended their functional structure [3], and later improvements by Bag and Samanta [4,5] laid a deeper foundation for fuzzy functional analysis, especially for bounded linear operators and fixed point theory. More recently, Alegre and Romaguera revisited the Hahn–Banach extension theorem in fuzzy normed spaces, further enriching the theory of fuzzy functional analysis [6].

In parallel, Atanassov introduced intuitionistic fuzzy sets, enhancing classical fuzzy set theory by associating each element with both a degree of membership and a degree of non-membership. This dualistic layer of uncertainty led Park [7] to define intuitionistic fuzzy metric spaces and Saadati and Park [8] to develop intuitionistic fuzzy normed spaces. These developments opened new directions for convergence structures, fixed point theory, and fuzzy topology [9,10,11]. More recently, Alegre and Romaguera introduced fuzzy quasi-norms [12], relaxing triangle inequality or symmetry conditions and allowing asymmetric or paratopological models; their subsequent work [13] established results such as the uniform boundedness theorem in this setting. Wu and Li [14] extended classical theorems, including the open mapping and closed graph theorems, into fuzzy quasi-normed spaces, while related studies on fuzzy inner product spaces and fuzzy norm functions [15] illustrate the continuing expansion of fuzzy functional analysis.

Although intuitionistic fuzzy normed spaces and fuzzy quasi-normed spaces are well studied, the integration of fuzzy logic into higher-dimensional normed structures, such as 2-normed spaces, remains underexplored. Classical 2-normed spaces, introduced by Gähler, generalize the concept of a norm by measuring the magnitude of a pair of vectors instead of a single vector, providing a framework for multilinear mappings, geometry, and dependence relations.

This work introduces and develops intuitionistic fuzzy 2-normed spaces (IF2NS). Unlike intuitionistic fuzzy normed spaces or fuzzy quasi-normed spaces, an IF2NS employs two variable intuitionistic fuzzy mappings that simultaneously track degrees of membership and non-membership. This dual two-variable structure captures hesitation and ambiguity beyond the reach of classical 2-norms or their fuzzy extensions, offering a clearer representation of uncertainty and stronger tools for analyzing pairwise dependence. A brief comparison with hesitant fuzzy 2-normed spaces is provided later to delineate domain differences.

Within this framework we introduce fuzzy open balls and examine the convergence of sequences, including left and right N Cauchy sequences with respect to the topologies ${\tau }_N$ and ${\tau }_{N^{-1}}$. We establish fuzzy analogues of fundamental functional analytic theorems, notably the open mapping and closed graph theorems, and we systematically compare these results with their classical and fuzzy counterparts. The intrinsic symmetry of the 2-norm and the intuitionistic fuzzy pair $(A,B)$, satisfying $A(x,y,t)=A(y,x,t)$ and $B(x,y,t)=B(y,x,t)$, naturally supports these generalizations.

By capturing dual uncertainty in a pairwise setting, IF2NS provides a tool for modeling complex data in areas such as multi-criteria decision making, information fusion, and uncertain network analysis, where relationships among pairs of variables are inherently fuzzy. This study was inspired by recent advances in intuitionistic fuzzy quasi-normed theory and by structural explorations in fuzzy operator theory and fuzzy topological vector spaces by several authors; our aim is to clarify the relationship between higher order fuzzy norms and existing frameworks and to stimulate further research in both theory and applications.

2. Preliminaries

This section lays out the fundamental notions necessary for constructing the theory of intuitionistic fuzzy 2-normed spaces. We begin with continuous t-norms and t-conorms, then recall the definitions of fuzzy sets, intuitionistic fuzzy sets, quasi-normed structures, classical 2-normed spaces, and their extensions to the intuitionistic fuzzy context. A brief reminder of the basic definition of a topology is included so that the induced structures can be described precisely.

Definition 1

([2]). A mapping $\text{ffi}:[0,1]\times [0,1]\to [0,1]$ is called a continuous t-norm provided it satisfies the following

(a)

$s\text{ffi}t=t\text{ffi}s$ for all $s,t\in [0,1]$.

(b)

$s\text{ffi}\left(t\text{ffi}u\right)=\left(s\text{ffi}t\right)\text{ffi}u$ for all $s,t,u\in [0,1]$.

(c)

$s\text{ffi}t\le u\text{ffi}d$ whenever $s\le u$ and $t\le d$.

(d)

$s\text{ffi}1=s$ for all $s\in [0,1]$.

(e)

$\text{ffi}$ is continuous on ${[0,1]}^2$.

Definition 2

([2]). A function $\circ :[0,1]\times [0,1]\to [0,1]$ is a continuous t-conorm if it satisfies the following:

(a)

$s\circ t=t\circ s$ for all $s,t\in [0,1]$.

(b)

$s\circ (t\circ u)=(s\circ t)\circ u$ for all $s,t,u\in [0,1]$.

(c)

$s\circ t\le u\circ d$ whenever $s\le u$ and $t\le d$.

(d)

$s\circ 0=s$ for all $s\in [0,1]$.

(e)

∘ is continuous on ${[0,1]}^2$.

Example 1.

The standard pair of operations

$$★(s,t):=min\{s,t\},\circ (s,t):=max\{s,t\},$$

define a continuous t-norm and a continuous t-conorm, often denoted respectively by $s\wedge t$ and $s\vee t$.

Example 2.

Let $V={\mathbb{R}}^2$. For $x=(x_1,x_2)$ and $y=(y_1,y_2)$ define

$$A(x,y,t):={\displaystyle \frac{t}{t+|x_1{y}_2-x_2{y}_1|}},B(x,y,t):={\displaystyle \frac{|x_1{y}_2-x_2{y}_1|}{t+|x_1{y}_2-x_2{y}_1|}},$$

for every $t>0$.

Take the t-norm and t-conorm as

$$\ast (a,b):=ab(\mathit{product}t-\mathit{norm}),\circ (a,b):=a+b-ab(\mathit{probabilistic}t-\mathit{conorm}).$$

Then the pair $(A,B)$ together with $(\ast ,\circ )$ satisfies all the axioms of an intuitionistic fuzzy 2-normed space: symmetry in $(x,y)$, homogeneity, the convolution properties with respect to ∗ and ∘, and the limiting conditions ${lim}_{t\to \infty }A(x,y,t)=1$ and ${lim}_{t\to 0^{+}}B(x,y,t)=1$. Hence $({\mathbb{R}}^2,A,B,\ast ,\circ )$ is an IF2NS whose t-norm and t-conorm are not the classical min and max.

Proposition 1

([12]). Let ★ and ∘ be continuous t-norm and t-conorm on $[0,1]$. From this, the following can be expressed:

(i)

If $0<d_2<d_1<1$, there exist $d_3,d_4\in (0,1)$ such that $d_1★d_3\ge d_2$ and $d_1\ge d_4\circ d_2$.

(ii)

For every $d\in (0,1)$ there exist $u,v\in (0,1)$ such that $u★u\ge d$ and $v\circ v\le d$.

Proof. 

(i) Fix $d_1\in (0,1)$ and consider $f\left(t\right):=d_1★t$. The map f is continuous and nondecreasing with $f\left(0\right)=0$ and $f\left(1\right)=d_1$. If $0<d_2<d_1$, by the intermediate value theorem, there exists $d_3\in (0,1)$ with $f\left(d_3\right)=d_2$; hence $d_1★d_3=d_2\ge d_2$.

For the t-conorm part, fix $d_2\in (0,1)$ and consider $g\left(t\right):=t\circ d_2$. Then g is continuous and nondecreasing, and $g\left(0\right)=d_2$ and $g\left(1\right)=1$. Since $d_2<d_1<1$, there exists $d_4\in (0,1)$ such that $g\left(d_4\right)=d_1$, i.e., $d_4\circ d_2=d_1\le d_1$, proving $d_1\ge d_4\circ d_2$.

(ii) The maps $h\left(t\right):=t★t$ and $k\left(t\right):=t\circ t$ are continuous and nondecreasing on $[0,1]$ with $h\left(0\right)=0$, $h\left(1\right)=1$ and $k\left(0\right)=0$, $k\left(1\right)=1$. Given $d\in (0,1)$, the intermediate value theorem yields $u,v\in (0,1)$ such that $h\left(u\right)=d$ and $k\left(v\right)=d$, which imply $u★u=d\ge d$ and $v\circ v=d\le d$. □

Definition 3

([7]). An intuitionistic fuzzy set A over a set X is the collection

$$A=\{(x,{\mu }_A\left(x\right),{\nu }_A\left(x\right))\mid x\in X\},$$

where ${\mu }_A:X\to [0,1]$ and ${\nu }_A:X\to [0,1]$ satisfy

$$0\le {\mu }_A\left(x\right)+{\nu }_A\left(x\right)\le 1\mathrm{for}\mathrm{all}x\in X.$$

Here ${\mu }_A\left(x\right)$ is the degree of membership, ${\nu }_A\left(x\right)$ the degree of non-membership, and ${\pi }_A\left(x\right)=1-{\mu }_A\left(x\right)-{\nu }_A\left(x\right)$ the hesitation margin.

Definition 4.

A topology on a set X is a collection $\mathcal{T}$ of subsets of X containing ⌀ and X, closed under arbitrary unions and finite intersections. The pair $(X,\mathcal{T})$ is then a topological space.

Definition 5.

Let V be an intuitionistic fuzzy 2-normed space. For $x\in V$, $r\in (0,1)$, and $t>0$, define the fuzzy open ball

$$B_A(x,r,t)=\{y\in V:A(x-y,z,t)>1-r,B(x-y,z,t)<r\mathrm{for}\mathrm{all}z\in V\setminus \{\mathrm{\Theta }\}\}.$$

The collection of all sets $U\subset V$ such that for every $x\in U$, there exists $r\in (0,1)$ and $t>0$ with $B_A(x,r,t)\subset U$ forming a topology on V, denoted as ${\tau }_A$. This is called the topology generated by the intuitionistic fuzzy 2-norm.

Definition 6.

Let X be a real vector space and $Q:X\to [0,\infty )$ a mapping. The pair $(X,Q)$ is a quasi-normed space if the following holds:

(i)

$Q\left(x\right)=Q(-x)=0$ implies $x=0$.

(ii)

$Q\left(\alpha x\right)=\alpha Q\left(x\right)$ for all scalars $\alpha \ge 0$.

(iii)

$Q(x+y)\le Q\left(x\right)+Q\left(y\right)$ for all $x,y\in X$.

Definition 7.

A 2-normed space is a pair $(X,\parallel ·,·\parallel )$ where X is a vector space with $dimX\ge 2$ and $\parallel ·,·\parallel :X\times X\to \mathbb{R}$ satisfying the following.

(i)

$\parallel x,y\parallel =0$ if x and y are linearly dependent.

(ii)

$\parallel x,y\parallel =\parallel y,x\parallel$ for all $x,y\in X$.

(iii)

$\parallel \alpha x,y\parallel =\left|\alpha \right|\parallel x,y\parallel$ for all scalars $\alpha \in \mathbb{R}$.

(iv)

$\parallel x+z,y\parallel \le \parallel x,y\parallel +\parallel z,y\parallel$ for all $x,y,z\in X$.

Definition 8.

An intuitionistic fuzzy 2-norm on a real vector space X is a pair of functions $F,G:X\times X\times {\mathbb{R}}^{+}\to [0,1]$ such that for all $x,y\in X$ and $t>0$.

(i)

$F(x,y,t)+G(x,y,t)\le 1$.

(ii)

$F(x,y,t)=F(y,x,t)$ and $G(x,y,t)=G(y,x,t)$.

(iii)

$F(\alpha x,y,t)=F(x,y,t/|\alpha \left|\right)$ and $G(\alpha x,y,t)=G(x,y,t/|\alpha \left|\right)$.

(iv)

$F(x+z,y,t+s)\ge F(x,y,t)\text{ffi}F(z,y,s)$.

(v)

$G(x+z,y,t+s)\le G(x,y,t)\circ G(z,y,s)$.

(vi)

${lim}_{t\to \infty }F(x,y,t)=1$ and ${lim}_{t\to 0^{+}}F(x,y,t)=0$.

(vii)

${lim}_{t\to \infty }G(x,y,t)=0$ and ${lim}_{t\to 0^{+}}G(x,y,t)=1$.

These preliminaries provide the basic framework needed for the development of intuitionistic fuzzy 2-normed spaces, which is explored in detail in the following section.

3. Main Results

In this section we develop the core structure of intuitionistic fuzzy 2-normed spaces and establish the basic topological tools needed for later functional analytic results. The following definitions incorporate consistent notation and slightly expanded explanations for clarity:

Definition 9.

Let V be a real vector space with $dimV\ge 2$, and let ∗ and ∘ denote continuous t-norm and t-conorm operations. Suppose there exist two functions $A,B:V\times V\times (0,\infty )\to [0,1]$. The structure $(V,A,B,\ast ,\circ )$ is called an intuitionistic fuzzy 2-normed space (IF2NS) if for all $x,y,z\in V$, $t,s>0$, and all nonzero scalars $\alpha \in \mathbb{R}\setminus \left\{0\right\}$:

1.

$A(x,y,t)+B(x,y,t)\le 1$.

2.

$A(x,y,t)=A(y,x,t)$ and $B(x,y,t)=B(y,x,t)$.

3.

$A(x,y,t)=1$ if x and y are linearly dependent, and $B(x,y,t)=0$ if x and y are linearly dependent.

4.

$A(\alpha x,y,t)=A(x,y,t/|\alpha \left|\right)$ and $B(\alpha x,y,t)=B(x,y,t/|\alpha \left|\right)$.

5.

$A(x+z,y,t+s)\ge A(x,y,t)\ast A(z,y,s)$.

6.

$B(x+z,y,t+s)\le B(x,y,t)\circ B(z,y,s)$.

7.

${lim}_{t\to \infty }A(x,y,t)=1$ and ${lim}_{t\to \infty }B(x,y,t)=0$.

8.

for fixed $x,y\in V$, the function $t\mapsto A(x,y,t)$ is left continuous and $t\mapsto B(x,y,t)$ is right continuous.

Definition 10.

Let $(V,A,B,\ast ,\circ )$ be an IF2NS. For $x\in V$, $r\in (0,1)$ and $t>0$, the open ball centered at x with radius parameter r and scale t is

$$B_A(x,r,t)=\{y\in V:A(x-y,z,t)>1-r,B(x-y,z,t)<r\mathrm{for}\mathrm{all}z\in V\setminus \left\{0\right\}\}.$$

The family

$${\mathcal{B}}_A\left(x\right)=\{B_A(x,r,t):r\in (0,1),t>0\}$$

forms a neighborhood base at x and generates a $T_0$-topology ${\tau }_A$ on V. For the zero vector Θ we write

$${\mathcal{B}}_A\left(\mathrm{\Theta }\right)=\{B_A(\mathrm{\Theta },r,t):r\in (0,1),t>0\}.$$

Definition 11.

A sequence $\left\{x_n\right\}\subset V$ converges to $x\in V$ with respect to the topology ${\tau }_A$ if for every $z\in V\setminus \left\{0\right\}$ and each $t>0$,

$$\underset{n\to \infty }{lim}A(x_n-x,z,t)=1\mathit{and}\underset{n\to \infty }{lim}B(x_n-x,z,t)=0.$$

Definition 12.

A sequence $\left\{x_n\right\}\subset V$ is called left N-Cauchy if for every $ϵ>0$ and each $z\in V\setminus \left\{0\right\}$ there exist $N\in \mathbb{N}$ such that for all $m,n\ge N$,

$$A(x_m-x_n,z,t)>1-ϵ\mathit{and}B(x_m-x_n,z,t)<ϵ.$$

Definition 13.

An IF2NS $(V,A,B,\ast ,\circ )$ is left complete if every left N-Cauchy sequence converges to a point of V with respect to ${\tau }_A$.

Definition 14.

A subset $S\subset V$ is said to be of half second category if there exists a sequence $\left\{M_n\right\}$ of subsets of S such that $S={\bigcup }_{n=1}^{\infty }{M}_n$ and for some $k\in \mathbb{N}$,

$${int}_{{\tau }_A}\left(\overline{M_k}\right)\ne ⌀.$$

Example 3.

Let $V={\mathbb{R}}^2$. For $x=(x_1,x_2)$ and $y=(y_1,y_2)$ define

$$A(x,y,t)={\displaystyle \frac{t}{t+|x_1{y}_2-x_2{y}_1|}},B(x,y,t)={\displaystyle \frac{|x_1{y}_2-x_2{y}_1|}{t+|x_1{y}_2-x_2{y}_1|}},t>0.$$

Then $A(x,y,t)+B(x,y,t)=1$ for all $t>0$, and $A(x,y,t)=1$ if x and y are linearly dependent. With t-norm $\ast (a,b)=min\{a,b\}$ and t-conorm $\circ (a,b)=max\{a,b\}$, the structure $(V,A,B,\ast ,\circ )$ satisfies all conditions of Definition 9 and is an intuitionistic fuzzy 2-normed space.

Theorem 1.

Let $(V,A,B,\ast ,\circ )$ be an intuitionistic fuzzy 2-normed space. For each fixed $y\in V\setminus \left\{\mathrm{\Theta }\right\}$ and every $t>0$, the map

$$x⟼A(x,y,t)$$

is upper semicontinuous on $(V,{\tau }_A)$, while

$$x⟼B(x,y,t)$$

is lower semicontinuous on $(V,{\tau }_A)$.

Proof. 

Fix $y\in V\setminus \left\{\mathrm{\Theta }\right\}$ and $t>0$. We prove upper semicontinuity of $x\mapsto A(x,y,t)$ at an arbitrary $x_0\in V$; the proof for B is analogous with the roles of ∗ and ∘ reversed.

Let $\epsilon >0$. By left continuity of $u\mapsto A(x_0,y,u)$ at $u=t$, choose $s\in (0,t)$ such that

$$A(x_0,y,t-s)>A(x_0,y,t)-{\displaystyle \frac{\epsilon }{2}}.$$

Since ∗ is continuous and has 1 as a neutral element, there exists $\delta \in (0,1)$ such that for all $a\in [0,1]$,

$$a\ast (1-\delta )\ge a-{\displaystyle \frac{\epsilon }{2}}.$$

Consider the ${\tau }_A$-neighborhood of $x_0$

$$U:=\{x\in V:A(x-x_0,y,s)>1-\delta \mathrm{and}B(x-x_0,y,s)<\delta \}.$$

Note that U contains the basic ball $B_A(x_0,\delta ,s)$ and is therefore open in ${\tau }_A$. For any $x\in U$, write $x=x_0+h$. With axiom (v) of Definition 9 with $(x,z,t,s)=(x_0,h,t-s,s)$,

$$A(x,y,t)=A(x_0+h,y,(t-s)+s)\ge A(x_0,y,t-s)\ast A(h,y,s).$$

By the choices of s and $\delta$ and the estimate on ∗,

$$A(x,y,t)\ge \left(A(x_0,y,t)-\frac{\epsilon }{2}\right)\ast (1-\delta )\ge A(x_0,y,t)-\epsilon .$$

Thus $A(·,y,t)$ is upper semicontinuous at $x_0$.

For B, fix $\epsilon >0$. By right continuity of $u\mapsto B(x_0,y,u)$ at $u=t$, choose $s\in (0,t)$ such that

$$B(x_0,y,t-s)<B(x_0,y,t)+{\displaystyle \frac{\epsilon }{2}}.$$

Since ∘ is continuous and has 0 as neutral element, there exists $\delta \in (0,1)$ such that for all $b\in [0,1]$,

$$b\circ \delta \le b+{\displaystyle \frac{\epsilon }{2}}.$$

With the same neighborhood U as above, for any $x=x_0+h\in U$, axiom (vi) of Definition 9 gives

$$B(x,y,t)=B(x_0+h,y,(t-s)+s)\le B(x_0,y,t-s)\circ B(h,y,s)\le \left(B(x_0,y,t)+\frac{\epsilon }{2}\right)\circ \delta \le B(x_0,y,t)+\epsilon .$$

Hence $B(·,y,t)$ is lower semicontinuous at $x_0$. This completes the proof. □

Theorem 2.

Let $(V,A,B,\ast ,\circ )$ be an intuitionistic fuzzy 2-normed space, $M\subset V$, and $\lambda >0$, $t>0$. Then the following can be expressed:

(i)

${int}_{{\tau }_A}\left(\lambda M\right)=\lambda ·{int}_{{\tau }_A}\left(M\right)$.

(ii)

${\overline{\lambda M}}^{{\tau }_A}=\lambda ·{\overline{M}}^{{\tau }_A}$.

Proof. 

(i) Suppose $x\in {int}_{{\tau }_A}\left(M\right)$. Then there exist $r\in (0,1)$ and $t>0$ such that $B_A(x,r,t)\subset M$. For $u=\lambda x$ we have

$$\lambda ·B_A(x,r,t)\subset B_A(\lambda x,r,\lambda t)\subset \lambda M,$$

because for any $y\in B_A(x,r,t)$ and any $z\ne \mathrm{\Theta }$,

$$A(u-\lambda y,z,\lambda t)=A\left(\lambda \right(x-y),z,\lambda t)=A(x-y,z,t)>1-r,$$

and similarly $B(u-\lambda y,z,\lambda t)<r$, by axiom (iv). Hence $u\in {int}_{{\tau }_A}\left(\lambda M\right)$ and $\lambda ·{int}_{{\tau }_A}\left(M\right)\subset {int}_{{\tau }_A}\left(\lambda M\right)$.

Conversely, let $y\in {int}_{{\tau }_A}\left(\lambda M\right)$. Then there exist $r\in (0,1)$ and $t>0$ with $B_A(y,r,t)\subset \lambda M$. Write $y=\lambda x$. If $u\in B_A(x,r,t/\lambda )$, then for all $z\ne \mathrm{\Theta }$,

$$A(x-u,z,t/\lambda )>1-r⟹A(y-\lambda u,z,t)>1-r,$$

and similarly $B(y-\lambda u,z,t)<r$, so $\lambda u\in B_A(y,r,t)\subset \lambda M$; hence $u\in M$. Therefore $x\in {int}_{{\tau }_A}\left(M\right)$ and $y\in \lambda ·{int}_{{\tau }_A}\left(M\right)$, proving the reverse inclusion.

(ii) Let $x\in {\overline{M}}^{{\tau }_A}$. Then $B_A(x,r,t)\cap M\ne ⌀$ for all $r\in (0,1)$ and $t>0$. As above,

$$B_A(\lambda x,r,\lambda t)\cap \lambda M\ne ⌀,$$

so $\lambda x\in {\overline{\lambda M}}^{{\tau }_A}$ and $\lambda ·{\overline{M}}^{{\tau }_A}\subset {\overline{\lambda M}}^{{\tau }_A}$. The reverse inclusion follows by scaling back by ${\lambda }^{-1}$ and repeating the same argument. Hence ${\overline{\lambda M}}^{{\tau }_A}=\lambda ·{\overline{M}}^{{\tau }_A}$. □

Theorem 3.

Let $(V,A,B,\ast ,\circ )$ and $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be intuitionistic fuzzy 2-normed spaces. A linear mapping $T:V\to U$ is continuous on V if and only if it is continuous at the origin ${\mathrm{\Theta }}_V\in V$.

Proof. 

(⇒) If T is continuous on V then it is continuous at ${\mathrm{\Theta }}_V$ by definition.

(⇐) Assume T is continuous at ${\mathrm{\Theta }}_V$. Fix an arbitrary $x_0\in V$ and choose $r\in (0,1)$, $t>0$, and $z\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$. By continuity at the origin, there exist $r^{\prime }\in (0,1)$, $s>0$, and $w\in V\setminus \left\{{\mathrm{\Theta }}_V\right\}$ such that

$$y\in B_A({\mathrm{\Theta }}_V,r^{\prime },s)⟹A^{\prime }(T\left(y\right),z,t)>1-r,B^{\prime }(T\left(y\right),z,t)<r.$$

For $x\in V$ write $y=x-x_0$. Because T is linear,

$$T\left(x\right)-T\left(x_0\right)=T\left(y\right).$$

If $y\in B_A({\mathrm{\Theta }}_V,r^{\prime },s)$, then $x\in B_A(x_0,r^{\prime },s)$ and

$$A^{\prime }(T\left(x\right)-T\left(x_0\right),z,t)>1-r,B^{\prime }(T\left(x\right)-T\left(x_0\right),z,t)<r.$$

Hence

$$x\in B_A(x_0,r^{\prime },s)⟹T\left(x\right)\in B_{A^{\prime }}(T\left(x_0\right),r,t).$$

This shows T is continuous at $x_0$. Since $x_0$ is arbitrary, T is continuous on all of V. □

Theorem 4.

Let $(V,A,B,\ast ,\circ )$ and $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be intuitionistic fuzzy 2-normed spaces, with V left N-complete and U Hausdorff. For a linear map $T:V\to U$, the following are equivalent:

1.

T is continuous on V.

2.

the graph $\mathcal{G}\left(T\right):=\left\{\right(x,T\left(x\right)):x\in V\}$ is closed in the product topology ${\tau }_A\times {\tau }_{A^{\prime }}$.

Proof. 

($1\Rightarrow 2$) Assume T is continuous. Let $(x_n,T\left(x_n\right))\in \mathcal{G}\left(T\right)$ with $(x_n,T\left(x_n\right))\to (x,y)$ in ${\tau }_A\times {\tau }_{A^{\prime }}$. Then $x_n\to x$ in ${\tau }_A$; hence $T\left(x_n\right)\to T\left(x\right)$ in ${\tau }_{A^{\prime }}$. By uniqueness of limits in the Hausdorff space U, $y=T\left(x\right)$, so $(x,y)\in \mathcal{G}\left(T\right)$, and $\mathcal{G}\left(T\right)$ is closed.

($2\Rightarrow 1$) Assume $\mathcal{G}\left(T\right)$ is closed. By linearity, it suffices to show continuity at ${\mathrm{\Theta }}_V$. Let $r\in (0,1)$, $t>0$, and $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$ be fixed; we show there exist $r^{\prime }\in (0,1)$, $s>0$, and $w\in V\setminus \left\{{\mathrm{\Theta }}_V\right\}$ such that

$$x\in B_A({\mathrm{\Theta }}_V,r^{\prime },s)⟹T\left(x\right)\in B_{A^{\prime }}({\mathrm{\Theta }}_U,r,t),$$

i.e.,

$$A^{\prime }(T\left(x\right),z^{\prime },t)>1-r\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t)<r.$$

Suppose, towards a contradiction, that no such $(r^{\prime },s,w)$ exists. Then for the chosen $(r,t,z^{\prime })$, one can construct a sequence $\left(x_n\right)\subset V$ with $x_n\to {\mathrm{\Theta }}_V$ in ${\tau }_A$ but

$$A^{\prime }(T\left(x_n\right),z^{\prime },t)\le 1-r\mathrm{or}{B}^{\prime }(T\left(x_n\right),z^{\prime },t)\ge r$$

(1)

for all n. Set $y_n:=T\left(x_n\right)$. Since $x_n\to {\mathrm{\Theta }}_V$, we have $(x_n,y_n)\in \mathcal{G}\left(T\right)$ and $(x_n,{\mathrm{\Theta }}_U)\to ({\mathrm{\Theta }}_V,{\mathrm{\Theta }}_U)$ in ${\tau }_A\times {\tau }_{A^{\prime }}$. Consider the difference pairs

$$(x_n,y_n)-(x_n,{\mathrm{\Theta }}_U)=({\mathrm{\Theta }}_V,y_n).$$

By the product topology, $({\mathrm{\Theta }}_V,y_n)$ converges to $({\mathrm{\Theta }}_V,y)$ in ${\tau }_A\times {\tau }_{A^{\prime }}$ for any ${\tau }_{A^{\prime }}$-cluster point y of $\left(y_n\right)$ (if necessary, pass to a subsequence to ensure convergence in U, which is Hausdorff and first-countable under ${\tau }_{A^{\prime }}$ due to the countable base given by rational $(r,t)$). Since $\mathcal{G}\left(T\right)$ is closed and $({\mathrm{\Theta }}_V,y_n)\in \mathcal{G}\left(T\right)$ for all n, every such limit point must lie in $\mathcal{G}\left(T\right)$; hence $y=T\left({\mathrm{\Theta }}_V\right)={\mathrm{\Theta }}_U$. Therefore any convergent subsequence of $\left(y_n\right)$ must converge to ${\mathrm{\Theta }}_U$.

This forces $A^{\prime }(y_n,z^{\prime },t)\to 1$ and $B^{\prime }(y_n,z^{\prime },t)\to 0$ along every convergent subsequence, contradicting the uniform violation in (1). Hence our assumption was false, and there exist $r^{\prime },s,w$ as required; thus T is continuous at ${\mathrm{\Theta }}_V$, and by linearity, on all of V. □

Theorem 5.

Let $(V,A,B,\ast ,\circ )$ be a left N-complete intuitionistic fuzzy 2-normed space and let $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be a Hausdorff IF2NS. Suppose $\mathcal{F}\subset \mathcal{L}(V,U)$ is a family of linear operators such that for every $x\in V$, $t>0$, and $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$ there exists $r_x\in (0,1)$ with

$$A^{\prime }(T\left(x\right),z^{\prime },t)>1-r_x\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t)<r_x\mathrm{for}\mathrm{all}T\in \mathcal{F}.$$

Then there exist $r\in (0,1)$, $t_0>0$, and $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$ such that for every $T\in \mathcal{F}$ and every $x\in B_A({\mathrm{\Theta }}_V,r,t_0)$ one has

$$A^{\prime }(T\left(x\right),z^{\prime },t_0)>1-r\mathit{and}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)<r.$$

Equivalently, the family $\mathcal{F}$ is uniformly bounded on a ${\tau }_A$-neighborhood of the origin.

Proof. 

Fix $t_0>0$ and $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$ (these will be kept the same throughout the proof). For $n\in \mathbb{N}$ define

$$E_n:=\left\{x\in V:A^{\prime }(T\left(x\right),z^{\prime },t_0)>1-\frac{1}{n}\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)<\frac{1}{n}\mathrm{for}\mathrm{all}T\in \mathcal{F}\right\}.$$

We show the following:

Claim 1: ${\displaystyle V=\bigcup _{n=1}^{\infty }{E}_n}$. For a fixed $x\in V$, by the pointwise boundedness hypothesis, there exists $r_x\in (0,1)$ such that for all $T\in \mathcal{F}$,

$$A^{\prime }(T\left(x\right),z^{\prime },t_0)>1-r_x\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)<r_x.$$

Choose n with $1/n\le r_x$. Then $x\in E_n$. Hence ${\bigcup }_n{E}_n=V$.

Claim 2: Each $E_n$ is ${\tau }_A$-closed. Let $\left(x_k\right)\subset E_n$ with $x_k\to x$ in ${\tau }_A$. Fix any $T\in \mathcal{F}$. By linearity and continuity of T at ${\mathrm{\Theta }}_V$ implied by the theorem on continuity at zero ⇔ global continuity once we know T is continuous; however, we cannot assume that yet. Instead, we argue directly using the definitions of ${\tau }_A$ and the continuity properties in U: for each fixed T, the maps $x\mapsto A^{\prime }(T\left(x\right),z^{\prime },t_0)$ and $x\mapsto B^{\prime }(T\left(x\right),z^{\prime },t_0)$ are respectively upper and lower semicontinuous compositions of $x\mapsto T\left(x\right)$ (linear) with the semicontinuous maps from the theorem on semi-continuity (applied in U). Hence

$$\underset{k\to \infty }{lim\; sup}\left(1-A^{\prime }(T\left(x_k\right),z^{\prime },t_0)\right)\le 1-A^{\prime }(T\left(x\right),z^{\prime },t_0),\underset{k\to \infty }{lim\; inf}{B}^{\prime }(T\left(x_k\right),z^{\prime },t_0)\ge B^{\prime }(T\left(x\right),z^{\prime },t_0).$$

Since $x_k\in E_n$, we have $A^{\prime }(T\left(x_k\right),z^{\prime },t_0)>1-\frac{1}{n}$ and $B^{\prime }(T\left(x_k\right),z^{\prime },t_0)<\frac{1}{n}$ for all k. Taking limits yields $A^{\prime }(T\left(x\right),z^{\prime },t_0)\ge 1-\frac{1}{n}$ and $B^{\prime }(T\left(x\right),z^{\prime },t_0)\le \frac{1}{n}$. Because the inequalities are strict on a dense set and U is Hausdorff, we can keep them strict by slightly reducing $1/n$ if necessary; thus $x\in E_n$. Therefore $E_n$ is closed.

Since $(V,{\tau }_A)$ is left N-complete, it is a Baire space (the standard Baire-category argument carries over because left N-completeness yields completeness of the underlying uniform structure generated by the basic neighborhoods $B_A({\mathrm{\Theta }}_V,r,t)$). If, for contradiction, every $E_n$ had empty interior, then $V={\bigcup }_n{E}_n$ would be a countable union of nowhere-dense sets, contradicting the Baire theorem. Hence there exists N with ${int}_{{\tau }_A}\left(E_N\right)\ne ⌀$.

Therefore, there are $r\in (0,1)$ and $s>0$ such that the basic ball $B_A({\mathrm{\Theta }}_V,r,s)\subset E_N$. In particular, for all $x\in B_A({\mathrm{\Theta }}_V,r,s)$ and every $T\in \mathcal{F}$,

$$A^{\prime }(T\left(x\right),z^{\prime },t_0)>1-\frac{1}{N}\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)<\frac{1}{N}.$$

Renaming $\frac{1}{N}$ by r (or simply setting $r:=\frac{1}{N}$) gives the desired uniform bound on the neighborhood $B_A({\mathrm{\Theta }}_V,r,s)$, which completes the proof. □

Remark 1.

A convenient gauge associated with $(A^{\prime },B^{\prime })$ at fixed $(z^{\prime },t_0)$ is

$${\parallel u\parallel }_{A^{\prime },B^{\prime }}^{z^{\prime },t_0}:=inf\left\{\rho >0:A^{\prime }(u,z^{\prime },t_0)\ge 1-\rho \mathit{and}{B}^{\prime }(u,z^{\prime },t_0)\le \rho \right\}.$$

The conclusion of Theorem 5 is equivalent to the existence of $r\in (0,1)$, $s>0$, and $z^{\prime }\ne {\mathrm{\Theta }}_U$ such that ${sup}_{T\in \mathcal{F}}{\parallel T\left(x\right)\parallel }_{A^{\prime },B^{\prime }}^{z^{\prime },t_0}<\infty$ for all $x\in B_A({\mathrm{\Theta }}_V,r,s)$.

Theorem 6.

Let $(V,A,B,\ast ,\circ )$ and $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be intuitionistic fuzzy 2-normed spaces such that V is left N-complete and U is Hausdorff, and let $\mathcal{T}:V\to U$ be linear. Then the following are equivalent:

(i)

$\mathcal{T}$ is continuous on V.

(ii)

the graph $\mathcal{G}\left(\mathcal{T}\right)=\left\{\right(x,\mathcal{T}x):x\in V\}$ is closed in ${\tau }_A\times {\tau }_{A^{\prime }}$.

(iii)

whenever $x_n\to 0$ in ${\tau }_A$, one has $\mathcal{T}\left(x_n\right)\to 0$ in ${\tau }_{A^{\prime }}$ (sequential continuity at the origin).

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$: If $\mathcal{T}$ is continuous, then for any sequence $(x_n,\mathcal{T}{x}_n)\to (x,y)$ in ${\tau }_A\times {\tau }_{A^{\prime }}$, we have $x_n\to x$, and hence $\mathcal{T}\left(x_n\right)\to \mathcal{T}\left(x\right)$; by uniqueness of limits in the Hausdorff space, U, $y=\mathcal{T}\left(x\right)$, so $(x,y)\in \mathcal{G}\left(\mathcal{T}\right)$. Thus $\mathcal{G}\left(\mathcal{T}\right)$ is closed.

$\left(ii\right)\Rightarrow \left(i\right)$: By Theorem 4, in a left N-complete domain and Hausdorff codomain, a linear operator has a closed graph if and only if it is continuous. Hence $\mathcal{T}$ is continuous.

$\left(i\right)\Rightarrow \left(iii\right)$: If $\mathcal{T}$ is continuous at 0, then $x_n\to 0$ implies $\mathcal{T}\left(x_n\right)\to 0$ by definition.

$\left(iii\right)\Rightarrow \left(i\right)$: Let $x_0\in V$ and suppose $x_n\to x_0$ in ${\tau }_A$. Then $x_n-x_0\to 0$, and by sequential continuity at 0,

$$\mathcal{T}\left(x_n\right)-\mathcal{T}\left(x_0\right)=\mathcal{T}(x_n-x_0)⟶0$$

in ${\tau }_{A^{\prime }}$. Hence $\mathcal{T}\left(x_n\right)\to \mathcal{T}\left(x_0\right)$ for every convergent sequence $x_n\to x_0$. Since $\mathcal{T}$ is linear and sequentially continuous at every point, it is continuous on V (cf. the previously proved result that for linear maps; continuity on V is equivalent to continuity at ${\mathrm{\Theta }}_V$). □

Theorem 7.

Let $(V,A,B,\ast ,\circ )$ and $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be intuitionistic fuzzy 2-normed spaces with V left N-complete and U Hausdorff. Let $T:V\to U$ be a bijective linear map. The following statements are equivalent:

(i)

T is continuous.

(ii)

$T^{-1}$ is continuous.

(iii)

the graph $\mathcal{G}\left(T\right)=\left\{\right(x,T\left(x\right)):x\in V\}$ is closed in the product topology ${\tau }_A\times {\tau }_{A^{\prime }}$.

(iv)

T is a topological isomorphism between $(V,{\tau }_A)$ and $(U,{\tau }_{A^{\prime }})$.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$: Suppose T is continuous and bijective. Let $\left(y_n\right)\subset U$ with $y_n\to y$ in ${\tau }_{A^{\prime }}$ and set $x_n:=T^{-1}\left(y_n\right)$. Since $T\left(x_n\right)=y_n\to y$ and T is continuous and injective, $x_n$ converges to $x:=T^{-1}\left(y\right)$ in ${\tau }_A$. Hence $T^{-1}$ is continuous.

$\left(ii\right)\Rightarrow \left(iii\right)$: Assume $T^{-1}$ is continuous. If $(x_n,T\left(x_n\right))\to (x,y)$ in ${\tau }_A\times {\tau }_{A^{\prime }}$, then $T\left(x_n\right)\to y$ and $x_n\to x$. Continuity of $T^{-1}$ gives $x_n=T^{-1}\left(T\left(x_n\right)\right)\to T^{-1}\left(y\right)$. By uniqueness of limits in the Hausdorff space V, $x=T^{-1}\left(y\right)$, so $y=T\left(x\right)$ and $(x,y)\in \mathcal{G}\left(T\right)$. Thus $\mathcal{G}\left(T\right)$ is closed.

$\left(iii\right)\Rightarrow \left(iv\right)$: If the graph of T is closed, the closed graph theorem for IF2NS (Theorem 4) applies because V is left N-complete and U is Hausdorff. Hence T is continuous. Bijectivity of T and the argument of $\left(i\right)\Rightarrow \left(ii\right)$ yield continuity of $T^{-1}$. Therefore T is a homeomorphism, i.e., a topological isomorphism.

$\left(iv\right)\Rightarrow \left(i\right)$: A topological isomorphism is a homeomorphism by definition and is particularly continuous.

These implications complete the equivalence. □

Theorem 8.

Let $(V,A,B,\ast ,\circ )$ be an intuitionistic fuzzy 2-normed space. Assume the following:

(i)

The t-norm ∗ is continuous and Archimedean.

(ii)

For each $x\in V$ and every $z\in V\setminus \{\mathrm{\Theta }\}$, the functions $t\mapsto A(x,z,t)$ and $t\mapsto B(x,z,t)$ are continuous.

(iii)

Every basic ball $B_A(x,r,t)$ is convex, balanced, and absorbing.

(iv)

The space $(V,{\tau }_A)$ is $T_0$ and admits a countable neighborhood base at Θ.

Then ${\tau }_A$ is metrizable.

Proof. 

Fix a countable neighborhood base at the origin ${\mathcal{B}}_0=\{B_A(\mathrm{\Theta },r_n,t_n):n\in \mathbb{N}\}$ with $r_n,t_n\in (0,1]$ and $r_n\downarrow 0$, $t_n\downarrow 0$. For each n, set

$$U_n:=B_A(\mathrm{\Theta },r_n,t_n).$$

By (iii) each $U_n$ is convex, balanced, and absorbing. Define the (Minkowski) gauge of $U_n$ by

$${\mu }_n\left(x\right):=inf\{\lambda >0:x\in \lambda U_n\}\in [0,\infty ).$$

Standard properties of gauges for convex, balanced, absorbing sets imply the following: ${\mu }_n(\mathrm{\Theta })=0$, ${\mu }_n(-x)={\mu }_n\left(x\right)$, ${\mu }_n\left(\alpha x\right)=\left|\alpha \right|{\mu }_n\left(x\right)$, and ${\mu }_n(x+y)\le {\mu }_n\left(x\right)+{\mu }_n\left(y\right)$ for all $x,y\in V$ and $\alpha \in \mathbb{R}$. The absorbing property uses (iii); balancedness follows from the symmetry and scaling axiom $A(\alpha x,y,t)=A(x,y,t/|\alpha \left|\right)$ together with the monotonic dependence on t ensured by (ii); convexity is given.

Define a translation-invariant metric d on V by the usual countable family of gauges:

$$d(x,y):=\sum _{n=1}^{\infty }{2}^{-n}{\displaystyle \frac{{\mu }_n(x-y)}{1+{\mu }_n(x-y)}}.$$

Each summand takes values in $[0,2^{-n})$, so the series converges, and d is a metric (symmetry and triangle inequality follow from those of each ${\mu }_n$; $d(x,y)=0$ implies ${\mu }_n(x-y)=0$ for all n; hence $x-y\in {\bigcap }_n{U}_n$, which by $T_0$ yields $x=y$).

We show that d generates ${\tau }_A$. First, if $x_k\to x$ in d, then ${\mu }_n(x_k-x)\to 0$ for every n, so for each fixed n and large k, we have $x_k-x\in U_n=B_A(\mathrm{\Theta },r_n,t_n)$. Hence $x_k\to x$ in ${\tau }_A$. Conversely, if $x_k\to x$ in ${\tau }_A$, then for every n eventually $x_k-x\in U_n$, i.e., ${\mu }_n(x_k-x)\le 1$; hence each summand tends to 0, and therefore $d(x_k,x)\to 0$. Thus d and ${\tau }_A$ have the same convergent sequences; hence the same topology (first countability at $\mathrm{\Theta }$ from (iv) suffices to conclude the equality of the topologies).

Finally, the Archimedean property in (i) guarantees that the base $\left\{U_n\right\}$ can be chosen to be nested (after thinning if necessary): for any $0<r^{\prime }<r<1$, there exists m with $(1-r)=(1-r^{\prime })\ast \dots \ast (1-r^{\prime })$ (repeated m times), which ensures $B_A(\mathrm{\Theta },r^{\prime },t)\subset B_A(\mathrm{\Theta },r,mt)$ using axiom (v) and continuity, simplifying the construction of the gauges without changing the generated topology.

Therefore $(V,{\tau }_A)$ is metrizable. □

Theorem 9.

Let $(V,A,B,\ast ,\circ )$ be an intuitionistic fuzzy 2-normed space. Then there exists a left N-complete IF2NS $(\overline{V},\overline{A},\overline{B},\ast ,\circ )$ and a linear isometric embedding $J:V\to \overline{V}$ such that the following holds:

1.

$J\left(V\right)$ is dense in $(\overline{V},{\tau }_{\overline{A}})$.

2.

for all $x,y\in V$ and $t>0$.

$$\overline{A}(J\left(x\right),J\left(y\right),t)=A(x,y,t),\overline{B}(J\left(x\right),J\left(y\right),t)=B(x,y,t);$$

3.

$(\overline{A},\overline{B})$ extends $(A,B)$ naturally to equivalence classes of left N-Cauchy sequences in V.

Proof. 

Let $\mathcal{C}$ be the set of all left N-Cauchy sequences $\left\{x_n\right\}\subset V$, i.e.,

$$\underset{m,n\to \infty }{lim}A(x_m-x_n,z,t)=1,\underset{m,n\to \infty }{lim}B(x_m-x_n,z,t)=0$$

for every $z\in V\setminus \{\mathrm{\Theta }\}$ and $t>0$.

Define an equivalence relation ∼ on $\mathcal{C}$ by

$$\left\{x_n\right\}\sim \left\{y_n\right\}⟺\underset{n\to \infty }{lim}A(x_n-y_n,z,t)=1,\underset{n\to \infty }{lim}B(x_n-y_n,z,t)=0$$

for all $z\in V\setminus \{\mathrm{\Theta }\}$ and $t>0$.

Let $\overline{V}:=\mathcal{C}/\sim$ and write $\left[x_n\right]$ for the class of $\left\{x_n\right\}$. Define

$$\overline{A}(\left[x_n\right],\left[y_n\right],t):=\underset{n\to \infty }{lim}A(x_n,y_n,t),\overline{B}(\left[x_n\right],\left[y_n\right],t):=\underset{n\to \infty }{lim}B(x_n,y_n,t).$$

These limits exist and are independent of representatives because of the defining relations of ∼ and the N-Cauchy property.

Vector operations are given pointwise:

$$\left[x_n\right]+\left[y_n\right]:=[x_n+y_n],\alpha \left[x_n\right]:=\left[\alpha x_n\right],$$

which are well defined and make $\overline{V}$ a real vector space.

The pair $(\overline{A},\overline{B})$ satisfies all intuitionistic fuzzy 2-norm axioms: symmetry, homogeneity, subadditivity, and the limiting conditions passing to the limit from A and B because these are preserved under pointwise limits.

Define $J:V\to \overline{V}$ by $J\left(x\right)=\left[\right\{x\left\}\right]$, with the constant sequence at x. Linearity of J is immediate. For all $x,y\in V$ and $t>0$,

$$\overline{A}(J\left(x\right),J\left(y\right),t)=\underset{n\to \infty }{lim}A(x,y,t)=A(x,y,t),$$

$$\overline{B}(J\left(x\right),J\left(y\right),t)=\underset{n\to \infty }{lim}B(x,y,t)=B(x,y,t),$$

so J is an isometric embedding.

For density $J\left(V\right)$, given $\left[x_n\right]\in \overline{V}$, the sequence $J\left(x_n\right)\in J\left(V\right)$ satisfies

$$\overline{A}(J\left(x_n\right),\left[x_k\right],t)\to 1,\overline{B}(J\left(x_n\right),\left[x_k\right],t)\to 0$$

as $n,k\to \infty$, showing $J\left(x_n\right)\to \left[x_n\right]$ in ${\tau }_{\overline{A}}$.

For completeness, let ${\left\{\left[x_n^{\left(k\right)}\right]\right\}}_k$ be a left N-Cauchy sequence in $\overline{V}$. For each n the sequence ${\left\{x_n^{\left(k\right)}\right\}}_k$ is N-Cauchy in V. Choose a diagonal sequence $y_n:=x_n^{\left(n\right)}$. Then $\left[y_n\right]$ is well defined in $\overline{V}$ and

$$\overline{A}(\left[x_n^{\left(k\right)}\right],\left[y_n\right],t)\to 1,\overline{B}(\left[x_n^{\left(k\right)}\right],\left[y_n\right],t)\to 0,$$

showing convergence in $\overline{V}$.

Thus $(\overline{V},\overline{A},\overline{B},\ast ,\circ )$ is a left N-complete intuitionistic fuzzy 2-normed space, and J satisfies all required properties. □

Theorem 10.

Let $(V,A,B,\ast ,\circ )$ be a left N-complete intuitionistic fuzzy 2-normed space over $\mathbb{R}$ such that the following holds:

1.

Every fuzzy bounded linear functional $f:V\to \mathbb{R}$ is continuous with respect to ${\tau }_A$.

2.

The dual space $V^{\ast }$ of all continuous fuzzy linear functionals is itself an IF2NS under a dual pair $(A^{\ast },B^{\ast })$.

3.

The canonical map $\mathcal{J}:V\to V^{\ast \ast }$, $\mathcal{J}\left(x\right)\left(f\right)=f\left(x\right)$, is an isometric embedding.

Then V is reflexive in the fuzzy sense: $\mathcal{J}$ is surjective and $V\cong V^{\ast \ast }$ as intuitionistic fuzzy 2-normed spaces.

Proof. 

For $f\in V^{\ast }$ define the dual fuzzy 2-norms

$$A^{\ast }(f,z,t):=inf\left\{1-r:\left|f\right(x\left)\right|<r,A(x,z,t)>1-r\right\},B^{\ast }(f,z,t):=sup\left\{r:\left|f\right(x\left)\right|>r,B(x,z,t)<r\right\},$$

which by assumption (ii) satisfy the axioms of an intuitionistic fuzzy 2-norm.

The canonical map $\mathcal{J}:V\to V^{\ast \ast }$ is defined by

$$\mathcal{J}\left(x\right)\left(f\right)=f\left(x\right),f\in V^{\ast }.$$

Linearity is immediate:

$$\mathcal{J}(x+y)\left(f\right)=f(x+y)=f\left(x\right)+f\left(y\right)=\mathcal{J}\left(x\right)\left(f\right)+\mathcal{J}\left(y\right)\left(f\right),$$

Similarly, $\mathcal{J}\left(\alpha x\right)=\alpha \mathcal{J}\left(x\right)$.

Injectivity follows from the fuzzy Hahn–Banach separation: if $\mathcal{J}\left(x\right)=0$, then $f\left(x\right)=0$ for all $f\in V^{\ast }$, so $x=\mathrm{\Theta }$.

Define on $V^{\ast \ast }$ the induced fuzzy 2-norms

$$A^{\ast \ast }(\mathcal{J}\left(x\right),\mathcal{J}\left(y\right),t):=A(x,y,t),B^{\ast \ast }(\mathcal{J}\left(x\right),\mathcal{J}\left(y\right),t):=B(x,y,t).$$

Then $\mathcal{J}$ preserves the fuzzy 2-norm and is an isometric embedding.

To prove surjectivity, let $F\in V^{\ast \ast }$. By completeness of V, the dual $V^{\ast }$ is an IF2NS and the fuzzy Hahn–Banach extension theorem (cf. Saadati Park 2006) provides an $x_F\in V$ such that

$$F\left(f\right)=f\left(x_F\right)\mathrm{for}\mathrm{all}f\in V^{\ast }.$$

Thus $F=\mathcal{J}\left(x_F\right)$.

Consequently $\mathcal{J}$ is a bijective isometry of V onto $V^{\ast \ast }$, showing that V is reflexive as an intuitionistic fuzzy 2-normed space. □

Theorem 11.

Let $(V,A,B,\ast ,\circ )$ be a left N-complete intuitionistic fuzzy 2-normed space, and let $(U,A^{\prime },B^{\prime },{\ast }^{\prime },{\circ }^{\prime })$ be a Hausdorff IF2NS. Let $\mathcal{F}\subset \mathcal{L}(V,U)$ be a family of (fuzzy) linear operators $T:V\to U$, with each being continuous.

Assume pointwise fuzzy boundedness : for every $x\in V$, $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$, and $t_0>0$, there exists $r_x\in (0,1)$ such that

$$A^{\prime }(T\left(x\right),z^{\prime },t_0)>1-r_x,B^{\prime }(T\left(x\right),z^{\prime },t_0)<r_x\mathit{for}\mathit{all}T\in \mathcal{F}.$$

Then there exist $r\in (0,1)$, $t_0>0$, and a fuzzy neighborhood $B_A({\mathrm{\Theta }}_V,r,t_0)\subset V$ such that for all $x\in B_A({\mathrm{\Theta }}_V,r,t_0)$,

$$\underset{T\in \mathcal{F}}{sup}\left(1-A^{\prime }(T\left(x\right),z^{\prime },t_0)\right)<r,\underset{T\in \mathcal{F}}{sup}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)<r.$$

Proof. 

Fix any $z^{\prime }\in U\setminus \left\{{\mathrm{\Theta }}_U\right\}$ and $t_0>0$. For $n\in \mathbb{N}$ define

$$E_n:=\bigcap _{T\in \mathcal{F}}\left\{x\in V:1-A^{\prime }(T\left(x\right),z^{\prime },t_0)\le \frac{1}{n}\mathrm{and}{B}^{\prime }(T\left(x\right),z^{\prime },t_0)\le \frac{1}{n}\right\}.$$

Step 1: $E_n$ is ${\tau }_A$-closed. For fixed T, the map $u\mapsto A^{\prime }(u,z^{\prime },t_0)$ is upper semicontinuous, and $u\mapsto B^{\prime }(u,z^{\prime },t_0)$ is lower semicontinuous on U (Theorem on semicontinuity). Hence the sets

$$\{u\in U:A^{\prime }(u,z^{\prime },t_0)\ge 1-\frac{1}{n}\}\mathrm{and}\{u\in U:B^{\prime }(u,z^{\prime },t_0)\le \frac{1}{n}\}$$

are closed in U. Since $T:V\to U$ is continuous, their preimages under T are ${\tau }_A$-closed in V. Intersecting over $T\in \mathcal{F}$ preserves closedness, so each $E_n$ is closed.

Step 2: ${\bigcup }_{n=1}^{\infty }{E}_n=V$. Given $x\in V$, by pointwise fuzzy boundedness there exists $r_x\in (0,1)$ such that for all $T\in \mathcal{F}$,

$$1-A^{\prime }(T\left(x\right),z^{\prime },t_0)<r_x,B^{\prime }(T\left(x\right),z^{\prime },t_0)<r_x.$$

Choose n with $1/n\le r_x$. Then $x\in E_n$. Hence ${\bigcup }_n{E}_n=V$.

Step 3: Baire-category argument. Since $(V,{\tau }_A)$ is left N-complete, it is a Baire space (the completeness of the underlying uniform structure generated by the basic neighborhoods $B_A({\mathrm{\Theta }}_V,r,t)$ suffices). If every $E_n$ had an empty interior, $V={\bigcup }_n{E}_n$ would be meager, contradicting the Baire property. Thus there exists $N\in \mathbb{N}$ and a nonempty open set $W\subset E_N$.

By first countability at ${\mathrm{\Theta }}_V$, we can choose $r\in (0,1)$ and $t_0>0$ so that the basic ball $B_A({\mathrm{\Theta }}_V,r,t_0)\subset W\subset E_N$. Then, for all $x\in B_A({\mathrm{\Theta }}_V,r,t_0)$ and all $T\in \mathcal{F}$,

$$1-A^{\prime }(T\left(x\right),z^{\prime },t_0)\le {\displaystyle \frac{1}{N}},B^{\prime }(T\left(x\right),z^{\prime },t_0)\le {\displaystyle \frac{1}{N}}.$$

Renaming $1/N$ by r (or taking any $r\in (1/(N+1),1/N]$) yields the asserted uniform bounds. □

Theorem 12.

Let $(V,A,B,\ast ,\circ )$ be a real intuitionistic fuzzy 2-normed space and let $C\subset V$ be nonempty, convex, and ${\tau }_A$ closed. Suppose there exist $x_0\notin C$, $t_0>0$, and $z\in V\setminus \{\mathrm{\Theta }\}$ such that

$$\underset{y\in C}{sup}A(x_0-y,z,t_0)<1,\underset{y\in C}{inf}B(x_0-y,z,t_0)>0.$$

Then there exists a nonzero continuous fuzzy linear functional $f:V\to \mathbb{R}$ with

$$f\left(x_0\right)>\underset{y\in C}{sup}f\left(y\right).$$

Proof. 

Choose $0<\delta <min\left\{1-{sup}_{y\in C}A(x_0-y,z,t_0),{inf}_{y\in C}B(x_0-y,z,t_0)\right\}$. The basic ball

$$B_A(x_0,\delta ,t_0):=\left\{x\in V:A(x-x_0,z,t_0)>1-\delta ,B(x-x_0,z,t_0)<\delta \right\}$$

is a ${\tau }_A$ neighborhood of $x_0$ disjoint from C; hence $x_0\notin \overline{C}$.

Let

$$\mathcal{K}:=C+B_A(\mathrm{\Theta },\delta ,t_0)=\{c+u:c\in C,u\in B_A(\mathrm{\Theta },\delta ,t_0)\}.$$

Because C is convex and $B_A(\mathrm{\Theta },\delta ,t_0)$ is convex, balanced, and absorbing (properties of IF2NS balls), $\mathcal{K}$ is convex and absorbing. Moreover $x_0\notin \mathcal{K}$.

Define the Minkowski functional

$$p\left(x\right):=inf\{\lambda >0:x\in \lambda \mathcal{K}\}.$$

The set $\mathcal{K}$ being convex and absorbing implies that p is a sublinear functional (gauge) on V.

Consider $U:=span\left\{x_0\right\}$ and define $f_0:U\to \mathbb{R}$ by

$$f_0\left(\alpha x_0\right):=\alpha p\left(x_0\right),\alpha \in \mathbb{R}.$$

Clearly $f_0$ is linear and satisfies $f_0\left(u\right)\le p\left(u\right)$ for all $u\in U$.

By the fuzzy Hahn–Banach extension theorem (see Saadati Park 2006), there exists a linear functional $f:V\to \mathbb{R}$ such that

$$f\left(x\right)\le p\left(x\right)\forall x\in V,$$

and f extends $f_0$.

Because $p\left(x_0\right)>p\left(y\right)$ for every $y\in C$ (as $x_0\notin \mathcal{K}$), we have

$$f\left(x_0\right)=p\left(x_0\right)>\underset{y\in C}{sup}f\left(y\right).$$

Finally, f is ${\tau }_A$ continuous. Indeed, if $x_n\to x$ in ${\tau }_A$, then $x_n-x\to 0$, so $p(x_n-x)\to 0$ by the continuity of p inherited from the balanced, absorbing neighborhood system. Since $|f(x_n-x)|\le p(x_n-x)$, it follows that $f\left(x_n\right)\to f\left(x\right)$. Thus $f\in V^{\ast }$ and separates $x_0$ from C as required. □

Theorem 13.

Let $(V,A,B,\ast ,\circ )$ be a left N-complete intuitionistic fuzzy 2-normed space and let $V^{\ast }$ denote the space of all continuous fuzzy linear functionals on V. Endow $V^{\ast }$ with the weak∗ fuzzy topology ${\tau }_w^{\ast }$, i.e., the coarsest topology for which every evaluation map ${\mathrm{\Phi }}_x:V^{\ast }\to \mathbb{R}$, ${\mathrm{\Phi }}_x\left(f\right)=f\left(x\right)$, is continuous for each $x\in V$. Then the closed fuzzy unit ball

$$B_{A^{\ast }}(0,1)=\left\{f\in V^{\ast }:A^{\ast }(f,z,t)\ge 1-\epsilon ,B^{\ast }(f,z,t)\le \epsilon ,\forall z\in V\setminus \{\mathrm{\Theta }\},t>0\right\}$$

is compact in $(V^{\ast },{\tau }_w^{\ast })$.

Proof. 

If $f_1\ne f_2$, there exists $x_0\in V$ with $f_1\left(x_0\right)\ne f_2\left(x_0\right)$. Since ${\mathrm{\Phi }}_{x_0}$ is continuous by definition of ${\tau }_w^{\ast }$, the preimages of disjoint intervals about $f_1\left(x_0\right)$ and $f_2\left(x_0\right)$ separate $f_1$ and $f_2$. Hence $(V^{\ast },{\tau }_w^{\ast })$ is Hausdorff.

Fix $x\in V$. We show ${sup}_{f\in B_{A^{\ast }}(0,1)}\left|f\left(x\right)\right|<\infty$. By the IF2NS axioms for $(A,B)$ and the definition of $(A^{\ast },B^{\ast })$ (dual fuzzy 2-norm), the condition

$$A^{\ast }(f,z,t)\ge 1-\epsilon ,B^{\ast }(f,z,t)\le \epsilon (\forall z\ne \mathrm{\Theta },t>0)$$

means that whenever $A(u,z,t)>1-\epsilon$ and $B(u,z,t)<\epsilon$, one has $\left|f\right(u\left)\right|\le \epsilon$. Using homogeneity and subadditivity in the first argument (axioms (iii), (iv) for $A,B$), for arbitrary x, we can scale $x=\lambda u$ so that u falls inside a basic ball $B_A(\mathrm{\Theta },\epsilon ,t)$; then

$$\left|f\right(x\left)\right|=\left|\lambda \right|\left|f\right(u\left)\right|\le \left|\lambda \right|\epsilon .$$

Since the same scaling works uniformly over all $f\in B_{A^{\ast }}(0,1)$, it follows that there exists $M_x<\infty$ with $\left|f\left(x\right)\right|\le M_x$ for all $f\in B_{A^{\ast }}(0,1)$. (Equivalently, with the gauge ${\parallel f\parallel }_{A^{\ast },B^{\ast }}^{z,t}:=inf\{\rho >0:A^{\ast }(f,z,t)\ge 1-\rho ,B^{\ast }(f,z,t)\le \rho \}$, the unit ball implies a uniform bound $\left|f\left(x\right)\right|\le C_x{\parallel f\parallel }_{A^{\ast },B^{\ast }}^{z,t}\le C_x$.)

For each $x\in V$, define the compact interval $I_x:=[-M_x,M_x]$. Consider the map

$$\mathrm{\Psi }:B_{A^{\ast }}(0,1)⟶\prod _{x\in V}{I}_x,\mathrm{\Psi }\left(f\right):={\left(f\left(x\right)\right)}_{x\in V}.$$

By definition of ${\tau }_w^{\ast }$, $\mathrm{\Psi }$ is continuous (all coordinates are the evaluation maps ${\mathrm{\Phi }}_x$). It is also injective: if $\mathrm{\Psi }\left(f_1\right)=\mathrm{\Psi }\left(f_2\right)$, then $f_1\left(x\right)=f_2\left(x\right)$ for all x; hence $f_1=f_2$.

The product ${\prod }_{x\in V}{I}_x$ is compact by the Tychonoff theorem. It remains to show that $\mathrm{\Psi }\left(B_{A^{\ast }}(0,1)\right)$ is closed in the product topology. Let ${\left(f_{\alpha }\right)}_{\alpha }$ be a net in $B_{A^{\ast }}(0,1)$ such that $\mathrm{\Psi }\left(f_{\alpha }\right)$ converges pointwise to a function $\phi ={\left(\phi \left(x\right)\right)}_{x\in V}$. Define $f:V\to \mathbb{R}$ by $f\left(x\right):=\phi \left(x\right)$. Linearity of f follows from pointwise limits of linear functionals. For continuity (i.e., $f\in V^{\ast }$), use the fact that each ${\mathrm{\Phi }}_x$ is ${\tau }_w^{\ast }$-continuous and that $B_{A^{\ast }}(0,1)$ is equi-fuzzy-bounded as shown above; hence for every fixed basic neighborhood $B_A(\mathrm{\Theta },r,t)$ in V, the family $\left\{f_{\alpha }\right\}$ is uniformly controlled on $B_A(\mathrm{\Theta },r,t)$, and this control passes to the limit, giving ${\tau }_A$ continuity of f. Moreover, the defining inequalities of $B_{A^{\ast }}(0,1)$ are preserved under pointwise limits because they are closed conditions with respect to the semicontinuity of $A^{\prime },B^{\prime }$ composed with evaluations, so $f\in B_{A^{\ast }}(0,1)$. Therefore $\mathrm{\Psi }\left(B_{A^{\ast }}(0,1)\right)$ is closed in ${\prod }_x{I}_x$.

Since $\mathrm{\Psi }$ is a continuous injection into a compact Hausdorff space with closed image, $\mathrm{\Psi }\left(B_{A^{\ast }}(0,1)\right)$ is compact. Thus $B_{A^{\ast }}(0,1)$ is compact in $(V^{\ast },{\tau }_w^{\ast })$. □

4. Conclusions

We introduced a unified framework of intuitionistic fuzzy 2-normed spaces (IF2NS) that simultaneously encodes membership and non-membership for pairs of vectors, thereby extending both classical 2-normed spaces and intuitionistic fuzzy normed spaces. Building from fuzzy open balls, convergence, and left N-Cauchy sequences, we developed the induced topology and showed semicontinuity of the basic maps $x\mapsto A(x,y,t)$ and $x\mapsto B(x,y,t)$. The intrinsic symmetry in the two=variable setting supports the core functional analytic results: continuity of linear maps from continuity at the origin, a closed graph theorem, an open mapping theorem, and a generalized isomorphism criterion. We established a metrization theorem under natural hypotheses (continuous Archimedean t-norm, continuity in t, convex balanced absorbing balls, and a countable base), a completion theorem via classes of left N-Cauchy sequences, and a uniform boundedness principle by a Baire-category argument adapted to IF2NS. On the dual side, we proved Hahn–Banach type separation and an Alaoglu type compactness result for the fuzzy weak∗ topology and derived a fuzzy reflexivity statement under appropriate dual-space assumptions. Illustrative examples included both the minimum/maximum pair and nontrivial choices such as the product t-norm with probabilistic sum t-conorm, clarifying how modeling choices affect topology and convergence.

This work highlights three contributions: a coherent two-variable intuitionistic fuzzy structure for uncertainty in pairwise geometry; systematic extensions of classical linear topological theorems to this setting; and explicit conditions ensuring metrizability, completeness, and compactness phenomena. Limitations include reliance on left N-completeness, Hausdorff codomains, and continuity/Archimedean assumptions on the t-operations; relaxing these may require alternative techniques or weaker conclusions. Future directions include operator theory on IF2NS (spectral properties, closed range and Fredholm alternatives), stability and approximation results, links with hesitant and type-2 fuzzy models, probabilistic/triangular norms beyond the Archimedean case, and applications to multi-criteria decision making and networked data where pairwise relations under uncertainty are fundamental.