On Structral Properties of Some Banach Space-Valued Schröder Sequence Spaces

Abstract

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties of vector-valued Schröder sequence spaces. Further, we show that these sequence spaces have a kind of Schauder basis. We also prove that ${\ell }_1\left(\mathcal{S},V\right)$ possesses the Dunford–Pettis property and demonstrate that ${\ell }_p\left(\mathcal{S},V\right)$ satisfies the approximation property for $1\le p<\infty$ under certain conditions and ${\ell }_{\infty }\left(\mathcal{S},V\right)$ has the Hahn–Banach extension property. Finally, we show that ${\ell }_2\left(\mathcal{S},V\right)$ has the Radon–Riesz property whenever V has it.

1. Introduction

The approximation property, alongside the Dunford–Pettis and Hahn–Banach extension properties, plays a pivotal role in Banach space theory. Although Banach spaces with a basis guarantee the approximation property (AP, for short), Per Enflo [1] proved that not all Banach spaces have this property. In general, vector-valued sequence and function spaces have no basis in the classical sense. However, in [2] a new kind Schauder basis definition introduced to investigate Banach spaces valued some normed spaces. In this context, some V-valued classical sequence spaces such as ${\ell }_{\infty }\left(V\right),c_0\left(V\right)$ and ${\ell }_p\left(V\right),$ $1\le p<\infty ,$ have this kind of basis where V is a Banach space. In this context, it can be concluded that some operator-valued or matrix-valued sequence spaces also have a basis. What was interesting in that study was that ${\ell }_{\infty }\left(V\right)$ and, of course, ${\ell }_{\infty }$ have a Schauder basis in the sense of [2], which is not in the form of a sequence but is in the form of a special net. This immediately leads us to the conclusion that such spaces have the AP. Using this basis definition, we also showed in [3] that some Banach space-valued Fibonacci sequence spaces have this type of basis. In more detail, V-valued Fibonacci sequence spaces ${\ell }_p\left(\mathcal{F},V\right)$ and ${\ell }_{\infty }\left(\mathcal{F},V\right),$ $1\le p<\infty ,$ which were introduced and investigated in [4,5], have this kind of basis and so they have the AP. Another important type of space is the $\mathcal{S}$ matrix created with the help of Schröder numbers and the group of Schröder sequence spaces created accordingly. Just like Fibonacci numbers, Schröder numbers are a group of numbers that have their own interesting properties and have some applications in some important areas of mathematics. Especially Schröder numbers have a very important position in combinatorics and number theory. In this context, by using the Schröder numbers, in [6,7], Cihat gave the matrix $\mathcal{S}$ and introduced the sequence spaces ${\ell }_p\left(\mathcal{S}\right),$ $1\le p<\infty ,$ and ${\ell }_{\infty }\left(\mathcal{S}\right)$. In addition, the characterizations of some matrix transformations related to these sequence spaces were given. We also know these spaces as classical Schröder sequence spaces. Some properties of compact operators on these spaces were also given in [6,7].

First of all let us now introduce the Schröder sequence $\left(S_n\right)$ explicitly in the form

$$1,2,6,22,90,394,1806,8558,41586,206098,1037718,\dots$$

with the associated recurrence relation

$$S_{n+1}=S_n+\sum _{k=0}^n{S}_k{S}_{n-k},\mathrm{for}n\ge 0\mathrm{and}{S}_0=1.$$

Furthermore, $S_n$ can also be represented using the functional formula

$$S_n=2{.}_2{F}_1(-n+1,n+2;2;-1)$$

where ${}_{2}F_1(-n+1,n+2;2;-1)$ is known as the hypergeometric function, which is defined as

$${}_{2}F_1(a,b;c;z)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}}{\displaystyle \frac{z^n}{n!}}.$$

At this point, we need to define the Pochhammer symbol ${\left(x\right)}_n=x(x+1)\dots (x+n-1)$ for $n\ge 1$, and ${\left(x\right)}_n=1$ for $n=0$. In [7], it presents a new matrix $\mathcal{S}=\left({\mathcal{S}}_{nk}\right)$ with entries

$${\mathcal{S}}_{nk}=\left\{\begin{array}{c}{\displaystyle \frac{S_k{S}_{n-k}}{S_{n+1}-S_n}},\\ 0,\end{array}\right.\begin{array}{c}if0\le k\le n\\ ifk>n\end{array}.$$

More clearly,

$$\mathcal{S}=\left[\begin{array}{ccccc}{\displaystyle \frac{S_0{S}_0}{S_1-S_0}}& 0& 0& 0& \dots \\ {\displaystyle \frac{S_0{S}_1}{S_2-S_1}}& {\displaystyle \frac{S_1{S}_0}{S_2-S_1}}& 0& 0& \dots \\ {\displaystyle \frac{S_0{S}_2}{S_3-S_2}}& {\displaystyle \frac{S_1{S}_1}{S_3-S_2}}& {\displaystyle \frac{S_2{S}_0}{S_3-S_2}}& 0& \dots \\ {\displaystyle \frac{S_0{S}_3}{S_4-S_3}}& {\displaystyle \frac{S_1{S}_2}{S_4-S_3}}& {\displaystyle \frac{S_2{S}_1}{S_4-S_3}}& {\displaystyle \frac{S_3{S}_0}{S_4-S_3}}& \dots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right].$$

We know from summability theory that infinite lower triangular matrices define bounded (continuous) linear operators among almost all classical sequence spaces (see [8]). By using this sub-triangular matrix $\mathcal{S}$ in [7], the sequence spaces ${\ell }_p\left(\mathcal{S}\right)$ $\left(\mathrm{for}1\le p<\infty \right)$ and ${\ell }_{\infty }\left(\mathcal{S}\right)$ are introduced, such that

$${\ell }_p\left(\mathcal{S}\right)=\left\{u=\left(u_n\right)\in w:\sum _{n=0}^{\infty }{\left|{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k\right|}^p<\infty \right\}$$

and

$${\ell }_{\infty }\left(\mathcal{S}\right)=\left\{u=\left(u_n\right)\in w:\underset{n}{sup}\left|{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k\right|<\infty \right\}.$$

Further, it is shown that ${\ell }_p\left(\mathcal{S}\right)$ and ${\ell }_{\infty }\left(\mathcal{S}\right)$ are BK spaces with the norms

$${∥u∥}_{{\ell }_p\left(\mathcal{S}\right)}={\left(\sum _{n=0}^{\infty }{\left|{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k\right|}^p\right)}^{1/p}$$

and

$${∥u∥}_{{\ell }_{\infty }\left(\mathcal{S}\right)}=\underset{n}{sup}\left|{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k\right|,$$

respectively. Although the constructions of these spaces are similar to the spaces ${\ell }_p\left(\mathcal{F}\right),$ $1\le p<\infty ,$ and ${\ell }_{\infty }\left(\mathcal{F}\right)$ given in [4,5], they have different properties because the Schröder numbers and Fibonacci numbers have different characteristics from each other in applications and nature. At least the characterization of matrix transformations on them is highly different. See [4,7] to see these differences. In parallel, the properties of V-valued Fibonacci sequence spaces, which we examined in [3], of course will be different from the spaces of V-valued Schröder sequence spaces ${\ell }_p\left(\mathcal{S},V\right)$ and ${\ell }_{\infty }\left(\mathcal{S},V\right)$, which we will define and examine below. Further, we will examine the operator-based properties of ${\ell }_p\left(\mathcal{S},V\right)$ and ${\ell }_{\infty }\left(\mathcal{S},V\right)$, which are clarified in the sequel.

It is an important advantage that bounded linear operators on Banach or Hilbert spaces have the property of being extended to a larger space. More precisely, for a Banach space X, given a closed subspace V of a Banach space U, and for a bounded linear operator $\Lambda :V\to X$, the space X is said to have the Hahn–Banach extension property if $\Lambda$ can be extended to a bounded linear operator $T:U\to X$ such that $∥\Lambda ∥=∥T∥$. We know that the classical sequence space ${\ell }_{\infty }$ has this property [9]. In [3] we proved that the Fibonacci sequence space ${\ell }_{\infty }\left(\mathcal{F},V\right)$ also has this property. We will see that the bounded V-valued Schröder sequence space ${\ell }_{\infty }\left(\mathcal{S},V\right)$ also has this property. Knowing that special operators have such extensions gives us the ability to obtain new operators suitable for special purposes. Furthermore, if we consider the AP, it is easier to study the spectral theory of compact linear operators defined on Banach spaces with the AP because compact operators on such spaces can be approximated step by step with bounded linear operators of finite rank. In this paper, we prove that under one condition the V-valued Schröder sequence space ${\ell }_p\left(\mathcal{S},V\right)$ has the AP.

Some further properties on Banach spaces such as Radon–Riesz and Dunford–Pettis allow us to better understand the algebraic and topological operator-based properties of the space. By these properties we can see some structural homogeneities and symmetries in these spaces. We show in this work that some V-valued Schröder sequence spaces also have these nice properties. The scalar-valued versions of these spaces were introduced and analyzed in [6,7].

Now, V-valued Schröder sequence spaces are defined by

$${\ell }_p\left(\mathcal{S},V\right)=\left\{u=\left(u_n\right)\in w\left(V\right):\sum _{n=0}^{\infty }{∥{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k∥}_V^p<\infty \right\}$$

and

$${\ell }_{\infty }\left(\mathcal{S},V\right)=\left\{u=\left(u_n\right)\in w\left(V\right):\underset{n}{sup}{∥{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k∥}_V<\infty \right\}$$

respectively. They are Banach spaces by the following corresponding norms:

$${∥u∥}_{{\ell }_p\left(\mathcal{S},V\right)}={\left(\sum _{n=0}^{\infty }{∥{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k∥}_V^p\right)}^{1/p}$$

and

$${∥u∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}=\underset{n}{sup}{∥{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{u}_k∥}_V,$$

as usual. For $V=\mathbb{K}$, for real or complex numbers, then ${\ell }_p\left(\mathcal{S},\mathbb{K}\right)={\ell }_p\left(\mathcal{S}\right)$ and ${\ell }_{\infty }\left(\mathcal{S},\mathbb{K}\right)={\ell }_{\infty }\left(\mathcal{S}\right).$ Further, we must notice that V-valued classical sequence spaces

$${\ell }_p\left(V\right)=\left\{u=\left(u_n\right)\in w\left(V\right):\sum _{n=0}^{\infty }{∥u_k∥}_V^p<\infty \right\}$$

and

$${\ell }_{\infty }\left(V\right)=\left\{u=\left(u_n\right)\in w\left(V\right):\underset{n}{sup}{∥u_k∥}_V<\infty \right\}$$

are Banach spaces with norms

$${∥u∥}_{{\ell }_p\left(V\right)}={\left(\sum _{n=0}^{\infty }{∥u_k∥}_V^p\right)}^{1/p}\mathrm{and}{∥u∥}_{{\ell }_{\infty }\left(V\right)}=\underset{n}{sup}{∥u_k∥}_V,$$

respectively.

We will prove that ${\ell }_1\left(\mathcal{S},V\right)$ possesses the Dunford–Pettis property, ${\ell }_2\left(\mathcal{S},V\right)$ has the Radon–Riesz property, and ${\ell }_p\left(\mathcal{S},V\right)$ has the approximation property for $1\le p<\infty$ under certain conditions. Further, we will see that ${\ell }_p\left(\mathcal{S},V\right)$ has an unconditional V-Schauder basis, while ${\ell }_{\infty }\left(\mathcal{S},V\right)$ has a conditional (unordered) V-Schauder basis, and will see that ${\ell }_{\infty }\left(\mathcal{S},V\right)$ obeys the Hahn–Banach extension property.

Now let us give some known definitions and theorems as a preparation for these properties. Assume U and V are Banach spaces. A linear operator $S:U\to V$ is called compact if it maps any bounded subset B in U to a relatively compact subset $S\left(B\right)$ in $V.$ Here $S\left(B\right)$ being relatively compact means $\overline{S\left(B\right)}$ is compact in $V.$ The set of all compact linear operators from U to V is written as $K(U,V),$ or $K\left(U\right)$ if $U=V.$ A compact operator between Banach spaces has closed range if and only if it has finite rank; that is, its range is finite dimensional. A Banach space U has, for every Banach space V, the set of all finite-rank bounded linear operators that is dense in $K(V,U)$ [10]. It is known that the spaces $c_0$ and ${\ell }_p,$ where $1\le p<\infty ,$ have the AP [11]. Any sequence $\left(x_n\right)$ in U is weakly convergent to x if $f\left(x_n\right)\to f\left(x\right)$ for all $f\in U^{*}$, the dual space of $U,$ and it is briefly written as $x_n\stackrel{w}{\to }x$ [11]. A linear operator $S:U\to V$ is called weakly compact if the image $S\left(B\right)$ of any bounded subset B in U is relatively weakly compact in $V.$ It is known that if T is a linear operator between U and V, then T is weakly compact if and only if, for any bounded sequence $\left(x_n\right)$ in $U,$ there exists its subsequence ${\left(x_{n_j}\right)}_{j=0}^{\infty }$ such that $\left(T{x}_{n_j}\right)$ is weakly convergent in V [11]. Let us now give another essential concept introduced by Hilbert [12]. A linear operator S from U into V is completely continuous, or a Dunford–Pettis operator, if S maps any weakly compact sets in U into compact sets in V.

Definition 1 

([11]). A Banach spaces U has the Dunford–Pettis property if U means that for every Banach spaces V, every weakly compact linear operator from U into V is a Dunford–Pettis operator.

It is well known that (see [11]) the classical Banach space ${\ell }_1$ has the Dunford–Pettis property.

Theorem 1 

(Phillips, R.S. [9]). Given a closed subspace $V$of a Banach space U and a bounded linear operator $\Lambda :V\to {\ell }_{\infty },$ Λ can be extended to a bounded linear operator $T:U\to {\ell }_{\infty }$ such that $∥\Lambda ∥=∥T∥.$

The operator $\Lambda$ described in the above theorem is sometimes called a Hahn–Banach operator and so it is said that ${\ell }_{\infty }$ enjoys the Hahn–Banach extension property. In this respect, we say a Banach space Z has the Hahn–Banach extension property if any closed subspace $V$of a Banach space U and a bounded linear operator $\Lambda :V\to Z,$ $\Lambda$ can be extended to a bounded linear operator $T:U\to Z$ such that $∥\Lambda ∥=∥T∥.$

Definition 2 

([2]). Consider Banach spaces U and V and let $\mathbb{A}$ be an index set, and $\mathcal{F}$ is the family of all finite subsets of $\mathbb{A}.$ A collection $\left\{{\eta }_a:a\in \mathbb{A}\right\}$ of continuous linear operators ${\eta }_a:V\to U$ is called a V-basis for Uif there exists a directed subset $\mathcal{D}$, by a relation $\ll ,$ of $\mathcal{F}$ and there exists a unique family $\left\{R_a:a\in \mathbb{A}\right\}$ of linear operators from U onto V such that the net $\left({\pi }_F\left(x\right):\mathcal{D}\right)$ converges to x in U for every $x\in U$, where each $F\in D$ and

$${\pi }_F\left(x\right)=\sum _{a\in F}\left({\eta }_a\circ R_a\right)\left(x\right).$$

Moreover, $\left\{{\eta }_a\right\}$ is called a V-Schauder basis for U whenever each operator $R_a$ is continuous. The V-basis in the definition is called unconditional if directed subset $\mathcal{D}$ is taken as whole of $\mathcal{F}$ with the inclusion relation $\subseteq .$

We call $\left\{R_a:a\in \mathbb{A}\right\}$ as the associate family of functions (A.F.F.) corresponding to V-basis $\left\{{\eta }_a:a\in \mathbb{A}\right\}.$

Given a V-basis $\left\{{\eta }_a:a\in \mathbb{A}\right\}$ for $U,$ the finite summation of function ${\pi }_F\left(x\right)$ induces an operator ${\pi }_F$ on U for each $F\in \mathcal{D}$. This operator is known as the F-projection on U relative to the V-basis.

Remark 1. 

Let V be a Banach space over the field $\mathbb{C}$ which has a Schauder basis $\left\{x_n\right\}$ in the classical sense. Then, the sequence $\left\{{\eta }_n\right\}$ of linear operators

$${\eta }_n:\mathbb{C}\to V:{\eta }_n\left(z\right)=z{x}_n$$

forms a $\mathbb{C}$-basis for V in the sense of above definition. In this case take $\mathbb{A}=\mathbb{N}$ and take

$$\mathcal{D}=\left\{\left\{1\right\},\left\{1,2\right\},\left\{1,2,3\right\},\dots \right\}$$

as a directed set in the poset $\mathcal{F}$ with the inclusion relation. Further, take $\left\{R_n\right\}$ as the sequence of coordinate functionals $\left(g_n\right)$ corresponding to the basis $\left\{x_n\right\}.$ Then we conclude that $\left({\pi }_F\left(x\right):\mathcal{D}\right)$ converges to x in U iff

$$\sum _{k=1}^n\left({\eta }_k\circ R_k\right)\left(x\right)=\sum _{k=1}^n{g}_k\left(x\right)x_k,$$

converges to $x={\displaystyle \sum _{n=1}^{\infty }}{g}_n\left(x\right)x_n.$

Theorem 2 

([2]). If a Banach space V possesses a Y-basis $\left\{{\eta }_a:a\in \mathbb{A}\right\},$ then V is separable iff the index set $\mathbb{A}$ is countable.

In this theorem, if the set $\mathbb{A}$ is not countable, the Banach space V may not be separable even if it has such a basis. We will see an example of this in the future.

2. Main Results

In this section, first we present key results concerning V-valued Schröder sequence spaces.

Theorem 3. 

It is given that a Banach space V the sequence space ${\ell }_p\left(\mathcal{S},V\right)$ has an unconditional V-Schauder basis.

Proof. 

Take $\mathbb{A}=\mathbb{N}$ and $\mathcal{D}=\mathcal{F}$, the family of all finite subsets of $\mathbb{N}$ with the inclusion relation. Consider embeddings

$$\begin{array}{ccc}\hfill I_n& :& V\to {\ell }_p\left(V\right)\hfill \\ \hfill I_n\left(z\right)& =& \left(0,\dots ,0,z,0,\dots \right).\hfill \end{array}$$

where z is in the $n.th$ place. Obviously each $I_n$ is a linear and bounded operator. Remember the Schröder matrix $\mathcal{S}$ above and observe easily that each $\mathcal{S}{I}_n$ is a bounded linear operator from V into ${\ell }_p\left(\mathcal{S},V\right).$ Now just the sequence

$$\left\{\mathcal{S}{I}_n:n\in \mathbb{N}\right\}$$

is, indeed, a V-Schauder basis for ${\ell }_p\left(\mathcal{S},V\right).$ Now let us prove this. First of all consider the coordinate projections

$$P_n:{\ell }_p\left(\mathcal{S},V\right)\to V;P_n\left(x\right)=x_n.$$

Since $\mathcal{D}$ is the collection of all finite subsets of $\mathbb{N}$, ordered by the inclusion relation ⊆, it is essential to prove that the net $\left({\pi }_F\left(x\right):\mathcal{D}\right)$ converges to x in ${\ell }_p\left(\mathcal{S},V\right)$.

$$\begin{array}{ccc}\hfill {\pi }_F\left(x\right)& =& \sum _{n\in F}\left(\left(\mathcal{S}{I}_n\right)\circ P_n\right)\left(x\right)\hfill \\ & =& \sum _{n\in F}\left(\mathcal{S}{I}_n\right)\left(x_n\right).\hfill \end{array}$$

It is clear that the convergence of the given net is equivalent to the convergence of the sequence of partial sums of the series ${\displaystyle \sum _{n=0}^{\infty }}\left(\mathcal{S}{I}_n\right)\left(x_n\right).$ Now, for any arbitrary $\epsilon >0$, we have to determine a finite subset $F_0=F_0\left(\epsilon \right)$ $\in \mathcal{D}$ such that for any finite set F containing $F_0,$ the desired property

$${∥x-{\pi }_F\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}\le \epsilon$$

is satisfied. As x is contained in ${\ell }_p\left(\mathcal{S},V\right)$, there necessarily exists a natural number $n_0\left(\epsilon \right)$ such that the series ${\displaystyle \sum _{n>n_0}^{\infty }}{∥{\left(\mathcal{S}x\right)}_n∥}_V^p$ is strictly smaller than $\epsilon .$ At this point, consider the set $F_0$ as

$$F_0=\left\{n\in \mathbb{N}:\sum _{n>n_0}^{\infty }{∥{\left(\mathcal{S}x\right)}_n∥}_V^p>\epsilon \right\}.$$

Then we get

$${∥x-{\pi }_F\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}={∥\left\{x_n:n\in \mathbb{N}\setminus F\right\}∥}_{{\ell }_p\left(\mathcal{S},V\right)}\le \epsilon ,$$

for each finite $F\supseteq F_0.$ This leads to the conclusion that $\left({\pi }_F\left(x\right):\mathcal{D}\right)$ converges to x in ${\ell }_p\left(\mathcal{S},V\right).$

Let us now verify that the sequence $\left\{P_n\right\}$ is uniquely defined. Suppose

$$\sum _{n\in \mathbb{N}}\left(\mathcal{S}{I}_n{P}_n\right)\left(x\right)=\sum _{n\in \mathbb{N}}\left(\mathcal{S}{I}_n{P}_n^{\prime }\right)\left(x\right)$$

and write

$${\pi }_F^{\circ }\left(x\right)=\sum _{n\in \mathbb{N}}\left(\mathcal{S}{I}_n\left(P_n-P_n^{\prime }\right)\right)\left(x\right),F\in \mathcal{D}.$$

Remember that

$${∥{\pi }_F^{\circ }\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}={\left(\sum _{n\in F}{∥\left(\mathcal{S}{I}_n\left(P_n-P_n^{\prime }\right)\right)\left(x\right)∥}^p\right)}^{1/p}$$

and

$${∥{\pi }_F^{\circ }\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}\le {∥{\pi }_G^{\circ }\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}$$

for $F\subseteq G.$ As $\left({\pi }_F\left(x\right):\mathcal{D}\right)$ exhibits convergence toward x in ${\ell }_p\left(\mathcal{S},V\right)$, we get

$$\underset{F\in \mathcal{D}}{lim}{∥{\pi }_F^{\circ }\left(x\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}=0.$$

From this observation, it follows that $\left(P_n-P_n^{\prime }\right)\left(x\right)=0$ for all n and for every $x\in {\ell }_p\left(\mathcal{S},V\right).$ Consequently, we obtain $P_n=P_n^{\prime }$ for each $n,$ ensuring the uniqueness of the basis.

Because the inequality ${∥x_n∥}_V\le {∥x∥}_{{\ell }_p\left(\mathcal{S},V\right)}$ is satisfied, each projection operator $P_n$ remains continuous. This confirms that $\left\{\mathcal{S}{I}_n:n\in \mathbb{N}\right\}$ indeed constitutes a V-Schauder basis for ${\ell }_p\left(\mathcal{S},V\right).$ Finally, the V-basis $\left\{\mathcal{S}{I}_n:n\in \mathbb{N}\right\}$ is unconditional since the directed set $\mathcal{D}$ is taken as whole of $\mathcal{F}$ with the inclusion relation $\subseteq .$ □

Theorem 4. 

The approximation property holds in ${\ell }_p\left(\mathcal{S},V\right)$ for all $1\le p<\infty$ if and only if the Banach space V has this property.

Proof. 

Suppose that $\Lambda$ is a compact linear operator from a Banach space V to ${\ell }_p\left(\mathcal{S},V\right).$ We seek a sequence $\left({\Lambda }_n\right)$ of bounded finite-rank linear operators from V into ${\ell }_p\left(\mathcal{S},V\right).$ Let us say early that ${\Lambda }_n$ will be determined as ${\mathcal{S}}^{-1}{A}_n$. Here, $\mathcal{S}$ is the Schröder matrix and its inverse exists. Now, due to the compactness of $\Lambda$, for any bounded sequence $\left(x_n\right)$ in $V,$ the sequence $\left(\Lambda x_n\right)$ contains a convergent subsequence ${\left(\Lambda x_{n_j}\right)}_{j=0}^{\infty }$ in ${\ell }_p\left(\mathcal{S},V\right)$. Further, for every $x\in V,$ $\Lambda x$ lies in ${\ell }_p\left(\mathcal{S},V\right)$ and explicitly

$$\begin{array}{ccc}\hfill {∥\Lambda x_{n_i}-\Lambda x_{n_j}∥}_{{\ell }_p\left(\mathcal{S},V\right)}^p& =& {∥\Lambda \left(x_{n_i}-x_{n_j}\right)∥}_{{\ell }_p\left(\mathcal{S},V\right)}^p\hfill \\ & =& \sum _{n=0}^{\infty }{∥{\displaystyle \frac{1}{S_{n+1}-S_n}}\sum _{k=0}^n{S}_k{S}_{n-k}{\left(\Lambda \left(x_{n_i}-x_{n_j}\right)\right)}_k∥}_V^p\hfill \\ & =& {∥\left(\mathcal{S}\Lambda \right)\left(x_{n_i}-x_{n_j}\right)∥}_{{\ell }_p\left(V\right)}^p.\hfill \end{array}$$

Just now we recall the classical Banach ${\ell }_p$ has the AP and with this conjecture we can say V has the AP if and only if ${\ell }_p\left(V\right)$ has it. Hence,

$${∥\left(\mathcal{S}\Lambda \right)\left(x_{n_i}-x_{n_j}\right)∥}_{{\ell }_p\left(V\right)}^p\to 0\mathrm{as}i,j\to \infty .$$

We conclude that the operator $\mathcal{S}\Lambda :V\to {\ell }_p\left(V\right)$ is compact. Since the matrix $\mathcal{S}$ is a bounded linear operator, $\mathcal{S}\Lambda$ is also bounded and linear. By the AP of ${\ell }_p\left(V\right)$, there exists a sequence of finite-rank bounded linear operators ${\left(A_m\right)}_{m=0}^{\infty }$ from $V$ to ${\ell }_p\left(V\right)$ such that $∥\mathcal{S}\Lambda -A_m∥\to 0$ as $m\to \infty .$ Thus, we obtain the desired sequence of finite-rank operators as ${\left({\mathcal{S}}^{-1}{A}_m\right)}_{m=0}^{\infty }$ which map $V$ into ${\ell }_p\left(\mathcal{S},V\right).$ Note that ${\mathcal{S}}^{-1}$ exists for the matrix $\mathcal{S}$ (see [7]), and it can easily be verified that each ${\mathcal{S}}^{-1}{A}_m$ is a bounded linear operator of finite rank. Further,

$$\begin{array}{ccc}\hfill ∥\Lambda -{\mathcal{S}}^{-1}{A}_m∥& =& \underset{∥x∥=1}{sup}{∥\left(\Lambda -{\mathcal{S}}^{-1}{A}_m\right)x∥}_{{\ell }_p\left(\mathcal{S},V\right)}\hfill \\ & =& \underset{∥x∥=1}{sup}{∥\Lambda x-\left({\mathcal{S}}^{-1}{A}_m\right)x∥}_{{\ell }_p\left(\mathcal{S},V\right)}^p\hfill \\ & =& \underset{∥x∥=1}{sup}{∥\mathcal{S}\Lambda x-\mathcal{S}\left({\mathcal{S}}^{-1}{A}_m\right)x∥}_{{\ell }_p\left(V\right)}^p\hfill \\ & =& \underset{∥x∥=1}{sup}{∥\left(\mathcal{S}\Lambda -A_m\right)x∥}_{{\ell }_p\left(V\right)}^p\hfill \\ & \to & 0\mathrm{as}m\to \infty .\hfill \end{array}$$

This completes the proof. □

Theorem 5. 

Suppose that V is a Banach space. Then ${\ell }_{\infty }\left(\mathcal{S},V\right)$ has a V-Schauder basis in the sense of Definition [2].

Proof. 

By Definition [2] let us take $\mathbb{A}=2^{\mathbb{N}}$, the power set of natural numbers, which is not a countable set. Let us describe D as the family of all finite partitions of $\mathbb{N}.$ Further, let $\mathcal{F}$ be the family of all finite subsets of $2^{\mathbb{N}}.$ Clearly D is a subfamily of $\mathcal{F}$ and let us impose following relation ⪯ on D: for ${\sigma }_1,{\sigma }_2\in D;$ ${\sigma }_1⪯{\sigma }_2$ if and only each ${\alpha }_j^{\prime }\in {\sigma }_2$ is included in some ${\alpha }_i\in {\sigma }_1$ with corresponding distinguished points $h_i\in {\alpha }_i$ and $h_j\in {\alpha }_j^{\prime },$ where ${\sigma }_1=\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)$ and ${\sigma }_2=\left({\alpha }_1^{\prime },{\alpha }_2^{\prime },\dots ,{\alpha }_m^{\prime }\right).$ Of course, each ${\alpha }_j^{\prime }$ and ${\alpha }_i$ are elements of $2^{\mathbb{N}}$. Now let us define mappings

$${\mu }_{\alpha }:V\to {\ell }_{\infty }\left(\mathcal{S},V\right),\mathrm{for}\alpha \in \mathbb{A},$$

such that for any $z\in V$

$${\mu }_{\alpha }\left(z\right)=\sum _{k\in \alpha }\left(\mathcal{S}{I}_k\right)\left(z\right)$$

again with the aid of embeddings $I_k\left(z\right)=\left(0,\dots ,0,z,0,0,\dots \right)$ into ${\ell }_{\infty }\left(V\right)$ such that z is in the kth place. Thus, the family $\left\{{\mu }_{\alpha }:\beta \in A\right\}$ is a $V-$ Schauder basis for ${\ell }_{\infty }\left(\mathcal{S},V\right).$ Let us prove this. Now, in the partially ordered set $\left(D,⪯\right)$ ${\sigma }_1⪯{\sigma }_2$ means that, partition ${\sigma }_2$ is finer than the partitions ${\sigma }_1$. Therefore, if ${\sigma }_1⪯{\sigma }_2$ and if $h_j\in {\alpha }_j^{\prime }$, then for some ${\alpha }_i\in {\sigma }_1$, $h_j\in {\alpha }_i$. For some $\alpha \in \mathbb{A}$ let us define $R_{\alpha }:{\ell }_{\infty }\left(\mathcal{S},V\right)\to V$, $R_{\alpha }\left(x\right)={\left({\mathcal{S}}^{-1}x\right)}_{h_{\alpha }}$where $h_{\alpha }$ is the distinguished point of $\alpha .$ Now consider the net $\left({\pi }_{\sigma }\left(x\right):\sigma \in D\right)$ such that

$${\pi }_{\sigma }\left(x\right)=\sum _{\alpha \in \sigma }\left({\mu }_{\alpha }\circ R_{\alpha }\right)\left(x\right)=\sum _{\alpha \in \sigma }{\mu }_{\alpha }\left({\left({\mathcal{S}}^{-1}x\right)}_{h_{\alpha }}\right).$$

Then

$$\begin{array}{ccc}\hfill {∥x-{\pi }_{\sigma }\left(x\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}& =& {∥x-\sum _{\alpha \in \sigma }{\mu }_{\alpha }\left({\left({\mathcal{S}}^{-1}x\right)}_{h_{\alpha }}\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& {∥x-\sum _{\alpha \in \sigma }\sum _{k\in \alpha }\mathcal{S}{I}_k{\left({\mathcal{S}}^{-1}x\right)}_{h_{\alpha }}∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& {∥x-\sum _{\alpha \in \sigma }{I}_{h_{\alpha }}\left(x\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}.\hfill \\ & =& ∥\left(x_1,x_2,\dots \right)-\left(0,\dots ,0,x_{h_1},0,\dots 0,x_{h_2},0,\dots ,0,x_{h_n},0,0,\dots \right)∥\hfill \end{array}$$

where if $\sigma =\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)$ and $h_i\in {\alpha }_i.$ From the definition, it is clearly seen that the convergence of the net ${∥x-{\pi }_{\sigma }\left(x\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}$ to zero means that the net $\left({\pi }_{\sigma }\left(x\right):\sigma \in D\right)$ converges to x in ${\ell }_{\infty }\left(\mathcal{S},V\right)$. The meaning of this net getting closer to x is that the $\sigma$ partitions become even thinner. In this case, the number of ${\alpha }_k$ elements in the partition goes towards infinity, and therefore the number of $h_k$’s goes towards infinity. This brings us that for every $\epsilon >0$ we can determine a ${\sigma }_0\in D$ such that for all ${\sigma }_0⪯\sigma$

$${∥x-{\pi }_{\sigma }\left(x\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}<\epsilon .$$

This completes the main part of the assertion. Further, the uniqueness is similar to that of ${\ell }_p\left(\mathcal{S},V\right).$ Finally, easily one can see that each $R_{\alpha }$ is continuous. □

Corollary 1. 

The family ${\mu }_{\alpha }:\mathbb{C}\to {\ell }_{\infty };$ ${\mu }_{\alpha }\left(t\right)=t{\chi }_{\alpha }$ is a $\mathbb{C}-$ Schauder basis for ${\ell }_{\infty },$ where ${\chi }_{\alpha }$ is the characteristic function of $\alpha \subset \mathbb{N}.$

It was shown in [2] that the $\mathbb{C}-$ Schauder basis for ${\ell }_{\infty }$ is not unconditional. Similarly, we can conclude that V-Schauder basis for ${\ell }_{\infty }\left(\mathcal{S},V\right)$ is not unconditional.

Remark 2. 

The basis we provided in the above theorem is an uncountable set. As a result, the space ${\ell }_{\infty }\left(\mathcal{S},V\right)$ is not a separable Banach space even though V is separable. Although more complicated than the convergence of sequences, the ability to present an approximation process for each element still provides a minor advantage.

Theorem 6. 

The Banach space ${\ell }_1\left(\mathcal{S},V\right)$ possesses the Dunford–Pettis property if and only if the space V also possesses it.

Proof. 

Suppose $\Lambda$ is a weakly compact linear operator acting from ${\ell }_1\left(\mathcal{S},V\right)$ into V. By composing it with ${\mathcal{S}}^{-1},$ we define $\Lambda {\mathcal{S}}^{-1}$, which is a bounded linear operator from ${\ell }_1\left(V\right)$ into $V.$ The weak compactness of $\Lambda {\mathcal{S}}^{-1}$ is guaranteed if and only if V is weakly compact. To prove this assertion, consider a bounded subset U of ${\ell }_1\left(V\right).$ Because ${\mathcal{S}}^{-1}$ is a bounded matrix operator, we conclude that ${\mathcal{S}}^{-1}\left(U\right)$ remains a bounded subset of ${\ell }_1\left(\mathcal{S},V\right).$ As a result, we see that

$$\Lambda \left({\mathcal{S}}^{-1}\left(U\right)\right)=\left(\Lambda {\mathcal{S}}^{-1}\right)\left(U\right)$$

is a relatively weakly compact subset in $V.$ Therefore, the operator $\Lambda {\mathcal{S}}^{-1}:{\ell }_1\left(V\right)\to V$ maintains weak compactness if and only if V does. Since ${\ell }_1\left(V\right)$ satisfies the Dunford–Pettis property precisely when V does, it follows that $\Lambda {\mathcal{S}}^{-1}$ is completely continuous. Finally, given that W is a weakly compact subset of ${\ell }_1\left(\mathcal{S},V\right)$, applying $\mathcal{S}$ ensures that $\mathcal{S}\left(W\right)$ remains a weakly compact subset of ${\ell }_1\left(V\right)$, and it follows that

$$\left(\Lambda {\mathcal{S}}^{-1}\right)\mathcal{S}\left(W\right)=\Lambda \left(W\right)$$

which is necessarily a compact subset of $V.$ □

Theorem 7. 

Assume V is a linear subspace of a Banach space $U,$ and let $\Lambda :V\to {\ell }_{\infty }\left(\mathcal{S},V\right)$ be a bounded linear operator. If Λ enjoys the Hahn–Banach extension property, then it is possible to construct a bounded linear operator $H:U\to {\ell }_{\infty }\left(\mathcal{S},V\right)$ that extends $\Lambda$with the property $∥H∥=∥\Lambda ∥.$

Proof. 

Take any bounded linear operator $\Lambda :V\to {\ell }_{\infty }\left(\mathcal{S},V\right).$ Since the Schröder matrix is bounded, it follows that $\mathcal{S}\Lambda :V\to {\ell }_{\infty }\left(V\right)$ defines a bounded linear operator. Moreover, because V satisfies the Hahn–Banach extension property, ${\ell }_{\infty }\left(V\right)$ inherits this property as well.

Given any $x\in V$ we have $\mathcal{S}\Lambda x\in {\ell }_{\infty }\left(V\right)$ and

$$\begin{array}{ccc}\hfill \mathcal{S}\Lambda x& =& \left({\left(\mathcal{S}\Lambda x\right)}_1,{\left(\mathcal{S}\Lambda x\right)}_2,\dots \right)\hfill \\ & =& \left(\left(P_1\mathcal{S}\Lambda \right)\left(x\right),\left(P_2\mathcal{S}\Lambda \right)\left(x\right),\dots \right).\hfill \end{array}$$

Each projection operator $P_n$ acts as a coordinate mapping from ${\ell }_{\infty }\left(V\right)$ onto V such that $P_n\left(x\right)=x_n.$ By applying the Hahn–Banach extension property of ${\ell }_{\infty }\left(V\right),$ it is possible to extend the bounded linear operator $\mathcal{S}\Lambda :V\to {\ell }_{\infty }\left(V\right)$ to a bounded linear operator $S:U\to {\ell }_{\infty }\left(V\right)$ while preserving its norm, meaning $∥S∥=∥\mathcal{S}\Lambda ∥.$ We introduce the operator H$:U$ → ${\ell }_{\infty }\left(\mathcal{S},V\right)$, ensuring that for any $x\in U,$ the mapping follows a precise definition

$$Hx=\left({\mathcal{S}}^{-1}S\right)\left(x\right).$$

The operator H is both well defined and linear because the operators S and ${\mathcal{S}}^{-1}$ satisfy these properties. As the relation

$$\begin{array}{ccc}\hfill {∥Hx∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}& =& {∥{\mathcal{S}}^{-1}\left(S\left(x\right)\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& {∥\mathcal{S}\left({\mathcal{S}}^{-1}\left(S\left(x\right)\right)\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & =& {∥S\left(x\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & \le & ∥S∥.∥x∥\hfill \end{array}$$

holds, we can conclude that H satisfies the boundedness property. For any $x\in V,$ we observe that

$$\begin{array}{ccc}\hfill {∥Hx∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}& =& {∥S\left(x\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & =& {∥\left(\mathcal{S}\Lambda \right)\left(x\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & =& {∥\Lambda x∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \end{array}$$

which establishes that H serves as an extension of $\Lambda .$Finally

$$\begin{array}{ccc}\hfill ∥H∥& =& \underset{{∥x∥}_U=1}{sup}{∥Hx∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& \underset{{∥x∥}_U=1}{sup}{∥{\mathcal{S}}^{-1}\left(S\left(x\right)\right)∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& \underset{{∥x∥}_U=1}{sup}{∥\mathcal{S}\left({\mathcal{S}}^{-1}\left(S\left(x\right)\right)\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & =& \underset{{∥x∥}_U=1}{sup}{∥S\left(x\right)∥}_{{\ell }_{\infty }\left(V\right)}\hfill \\ & =& \underset{{∥x∥}_U=1}{sup}{∥\Lambda x∥}_{{\ell }_{\infty }\left(\mathcal{S},V\right)}\hfill \\ & =& ∥\Lambda ∥.\hfill \end{array}$$

□

Let us now mention another elegance property of Banach spaces.

Definition 3 

([11]). A normed space X has the Radon–Riesz or Kadets–Klee properties if it satisfies the following condition: Whenever for any sequence $\left(x_n\right)$ and for an element$x$ in X such that $x_n\stackrel{w}{\to }x$ and $∥x_n∥\to ∥x∥$ implies $x_n\to x$ in $X.$

The proof of following lemma is straightforward.

Lemma 1. 

Given any Hilbert space $V,$ ${\ell }_2\left(\mathcal{S},V\right)$ is a Hilbert space by the inner product

$${〈u,v〉}_{{\ell }_2\left(\mathcal{S},V\right)}={〈\mathcal{S}u,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}=\sum _{k=1}^{\infty }{〈{\left(\mathcal{S}u\right)}_k,{\left(\mathcal{S}v\right)}_k〉}_V.$$

Theorem 8. 

For any Hilbert space V, ${\ell }_2\left(\mathcal{S},V\right)$ has the Radon–Riesz property if V has it.

Proof. 

Consider a sequence $\left(v_n\right)$ in ${\ell }_2\left(\mathcal{S},V\right)$ and an element u of the same space. Suppose that $v_n$ $\stackrel{w}{\to }$v in ${\ell }_2\left(\mathcal{S},V\right)$ and that the norms satisfy ${∥v_n∥}_{{\ell }_2\left(\mathcal{S},V\right)}\to {∥v∥}_{{\ell }_2\left(\mathcal{S},V\right)}.$ Our aim is to prove that $\left(v_n\right)$ is norm convergent to $v,$ meaning $v_n\to v$ in ${\ell }_2\left(\mathcal{S},V\right).$ By assumption, $v_n\stackrel{w}{\to }v$ implies that $f\left(v_n\right)\to f\left(v\right)$ for all functionals f in ${\ell }_2{\left(\mathcal{S},V\right)}^{*}.$ To finalize the proof, we will establish that ${∥v_n-v∥}_{{\ell }_2\left(\mathcal{S},V\right)}\to 0$.

$$\begin{array}{ccc}\hfill {∥v_n-v∥}_{{\ell }_2\left(\mathcal{S},V\right)}^2& =& {〈\mathcal{S}{v}_n-\mathcal{S}v,\mathcal{S}{v}_n-\mathcal{S}v〉}_{{\ell }_2\left(V\right)}\hfill \\ & =& {〈\mathcal{S}{v}_n,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}-{〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}\hfill \\ & & -{〈\mathcal{S}v,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}+{〈\mathcal{S}v,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}\hfill \\ & =& {∥\mathcal{S}{v}_n∥}_{{\ell }_2\left(V\right)}^2+{∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2-{〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}-{〈\mathcal{S}v,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}\hfill \end{array}$$

Let $y=\mathcal{S}v$ be an element of ${\ell }_2\left(V\right),$ naturally identified with ${\ell }_2{\left(V\right)}^{*},$and think of y as a functional and define the mapping $y\circ \mathcal{S}$ by the relation $\left(y\circ \mathcal{S}\right)v={〈\mathcal{S}v,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}.$ Exploiting the characteristics of the Schröder matrix $\mathcal{S}$, along with Riesz’s Theorem in the Hilbert space ${\ell }_2\left(V\right),$ we deduce that $y\circ \mathcal{S}$ is indeed a continuous linear functional on ${\ell }_2\left(\mathcal{S},V\right)$ and

$$\left(y\circ \mathcal{S}\right)v_n=y\left(\mathcal{S}{v}_n\right)={〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}.$$

Under the hypothesis $v_n\stackrel{w}{\to }v,$ we can deduce

$$\begin{array}{ccc}\hfill \left(y\circ \mathcal{S}\right)\left(v_n\right)& =& {〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}\hfill \\ & \to & {〈\mathcal{S}v,\mathcal{S}v〉}_{{\ell }_2\left(V\right)},\mathrm{as}n\to \infty ,\hfill \\ & =& \left(y\circ \mathcal{S}\right)\left(v\right)\hfill \\ & =& {∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2\hfill \end{array}$$

Taking a dual approach, we assign $y_n=\mathcal{S}{v}_n$, ensuring that each $y_n$ belongs to ${\ell }_2{\left(V\right)}^{*}={\ell }_2\left(V\right),$ for each $n$, and then

$$\left(y_n\circ \mathcal{S}\right)v=y_n\left(\mathcal{S}v\right)={〈\mathcal{S}v,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}.$$

As before, each $y_n\circ \mathcal{S}$ is a continuous linear functional acting on ${\ell }_2\left(\mathcal{S},V\right).$ Under the given assumption that $v_n\stackrel{w}{\to }v$, we deduce that

$$\begin{array}{ccc}\hfill \left(y_n\circ \mathcal{S}\right)\left(v\right)& =& y_n\left(\mathcal{S}v\right)\hfill \\ & =& {〈\mathcal{S}v,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}\hfill \\ & =& \overline{{〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}}\hfill \\ & \to & \overline{{〈\mathcal{S}v,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}},\mathrm{as}n\to \infty ,\hfill \\ & =& {∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2.\hfill \end{array}$$

By the hypothesis ${∥v_n∥}_{{\ell }_2\left(\mathcal{S},V\right)}\to {∥v∥}_{{\ell }_2\left(\mathcal{S},V\right)},$ we conclude that

$$\begin{array}{ccc}\hfill {∥v_n-v∥}_{{\ell }_2\left(\mathcal{S},V\right)}^2& =& {∥\mathcal{S}{v}_n∥}_{{\ell }_2\left(V\right)}^2+{∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2-{〈\mathcal{S}{v}_n,\mathcal{S}v〉}_{{\ell }_2\left(V\right)}-{〈\mathcal{S}v,\mathcal{S}{v}_n〉}_{{\ell }_2\left(V\right)}\hfill \\ & \to & {∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2+{∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2-{∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2-{∥\mathcal{S}v∥}_{{\ell }_2\left(V\right)}^2\hfill \\ & =& 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

□

3. Conclusions

The results in this paper can be much more useful if the Banach space V is chosen to be the Banach space of some types of matrices or the Banach space of some class of linear operators. For example, if the non-convergent sequences of operators of finite rank can be summed to an operator after applying the infinite matrix $\mathcal{S}$, this has important benefits. For this purpose we need to work in the appropriate space ${\ell }_p\left(\mathcal{S},V\right),$ especially in ${\ell }_2\left(\mathcal{S},V\right),$ and need to use the norm of this space. Furthermore, by using the concept of the V-Schauder basis, for example, if we choose the Banach space V of the matrices, we have the advantage that we can approach each matrix step by step by means of basis vectors consisting of sequences of such matrices. Of course, in this case, the terms of the V-Schauder basis will be matrices.