# On a Characterization of Convergence in Banach Spaces with a Schauder Basis

Authors: Marat V. Markin; Olivia B. Soghomonian
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

We extend the well-known characterizations of convergence in the spaceslp (1≤p<∞) of p-summable sequences and c0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences.
“The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications.”Jacob T. Schwartz


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## Body

## 1. Introduction

In normed vector spaces of sequences, termwise convergence, being a necessary condition for convergence of a sequence (of sequences), falls short of being characteristic (see, e.g., [1]). Thus, the natural question is as follows: what conditions are required to be, along with termwise convergence, necessary and sufficient for convergence of a sequence in such spaces?It turns out that, in the Banach spaceslp (1≤p<∞) of p-summable sequences with p-norm,(1)x≔xkk∈ℕ↦xp≔∑k=1∞xkp1/p,(ℕ≔1,2,… is the set of natural numbers) and c0 of vanishing sequences with ∞-norm,(2)x≔xkk∈ℕ↦x∞≔supk∈ℕxk,only one additional condition is needed. The following characterizations of convergence in the foregoing spaces are well known.Proposition 1 (characterization of convergence inlp (1≤p<∞)).
In the (real or complex) spacelp1≤p<∞,(3)xknk∈ℕ≕xn⟶x≔xkk∈ℕ,n⟶∞,iff(1)
∀k∈ℕ:xkn⟶xk, n⟶∞,(2)
∀ε>0∃K∈ℕ∀n∈ℕ:∑k=K+1∞xknp<ε.See, e.g., Proposition 2.16 in [2] and Proposition 2.17 in [1].Remarks 1.
(i)
Condition (1) is termwise convergence.(ii)
Condition (2) signifies the uniform convergence of the series,(4)∑k=1∞xknp,to their respective sums over n∈ℕ.Proposition 2 (characterization of convergence inc0).
In the (real or complex) spacec0,(5)xknk∈ℕ≕xn⟶x≔xkk∈ℕ,n⟶∞,iff(1)
∀k∈ℕ:xkn⟶xk, n⟶∞,(2)
∀ε>0∃K∈ℕ∀n∈ℕ:supk≥K+1xkn<ε.See, e.g., Proposition 2.15 in [2] and Proposition 2.16 in [1].Remarks 2.
(i)
Condition (1) is termwise convergence.(ii)
Condition (2) signifies the uniform convergence of the sequencesxknk∈ℕ to 0 over n∈ℕ.One cannot but notice that both characterizations share the same condition (1) and that condition (2) in each can be reformulated in the following equivalent form:(2C)∀ε>0∃K0∈ℕ∀K≥K0∀n∈ℕ:RKxn<ε,where⋅ stands for p-norm ⋅p (1≤p<∞) or ∞-norm, respectively, and the mapping RK:X⟶X, K∈ℕ, (X≔lp (1≤p<∞) or X≔c0) is defined as follows:(6)x≔xkk∈ℕ↦RKx≔0,…,0︸Kterms,xK+1,xK+2,…,K∈ℕ.Thus, we have the following combined characterization encompassing bothlp (1≤p<∞) and c0.Proposition 3 (combined characterization of convergence).
In the (real or complex) spaceX≔lp (1≤p<∞) or X≔c0,(7)xknk∈ℕ≕xn⟶x≔xkk∈ℕ,n⟶∞,iff(1)∀k∈ℕ:xkn⟶xk, n⟶∞,(2C)∀ε>0∃K0∈ℕ∀K≥K0∀n∈ℕ:RKxn<ε,where ⋅ stands for p-norm ⋅p (1≤p<∞) or ∞-norm, respectively, and the mapping RK:X⟶X, K∈ℕ, is defined by (6).In view of the fact that bothlp (1≤p<∞) and c0 are Banach spaces with a Schauder basis, our goal to show that a two-condition characterization of convergence, similar to the foregoing combined characterization, holds for all such spaces appears to be amply motivated. We establish a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the Banach space c of convergent sequences.


## 2. Preliminaries

Here, we briefly outline certain preliminaries essential for our discourse.Definition 1 (Schauder basis).
A Schauder basis (also a countable basis) of a (real or complex) Banach spaceX,⋅ is a countably infinite set enn∈ℕ in X such that(8)∀x∈X∃!ckxk∈ℕ∈Fℕ:x=∑k=1∞ckxek,F≔ℝorF≔ℂ the series called the Schauder expansion of x and the numbers ckx∈F, k∈ℕ, the coordinates of x relative to enn∈ℕ.See, e.g., [1–4].For an infinite-dimensional separable Hilbert spaceX,⋅,⋅,⋅ (⋅,⋅ stands for inner product and ⋅ for inner product norm), an orthonormal basis enn∈ℕ is a Schauder basis, and for an arbitrary x∈X,(9)x=∑k=1∞ckxek,withckx=x,ek,k∈ℕ,(see, e.g., [1, 2]).As we mention above, the sequence spaceslp (1≤p<∞), c0, and c are examples of Banach spaces with a Schauder basis. For lp (1≤p<∞) and c0, the standard Schauder basis is the set(10)en≔δnkk∈ℕn∈ℕ(δnk is the Kronecker delta) and for an arbitrary x≔xkk∈ℕ in the foregoing spaces,(11)x=∑k=1∞ckxek,withckx=xk,k∈ℕ,(see, e.g., [1–4]).For the Banach spacec of convergent sequences equipped with ∞-norm (see (2)), the standard Schauder basis is enn∈ℤ+ (ℤ+≔0,1,2,… is the set of nonnegative integers) with(12)e0≔1,1,1,…,and for an arbitrary x≔xkk∈ℕ∈c,(13)x=∑k=0∞ckxek,withc0x=limm⟶∞xm,ckx=xk−c0x,k∈ℕ,see, e.g., [1–4].Banach spaces with more sophisticated Schauder bases encompassLpa,b (1≤p<∞) and Ca,b−∞<a<b<∞ with ∞-norm,(14)Ca,b∋x↦x∞≔maxa≤t≤bxt,(see, e.g., [3, 4]).A Banach space with a Schauder basis is infinite-dimensional and separable (see, e.g., [1–3]). However, an infinite-dimensional separable Banach space need not have a Schauder basis (see [5]).The set ofF-termed sequences,(15)Y≔y≔ckk∈ℕ∈Fℕ∑k=1∞ckekconverges inX,with termwise linear operations and the norm,(16)Y∋y≔ckk∈ℕ↦yY≔supn∈ℕ∑k=1nckek,is a Banach space and the linear operator,(17)Y∋y≔ckk∈ℕ↦Ay≔∑k=1∞ckek∈X,is subject to the inverse mapping theorem (see, e.g., [1–3, 6]). The boundedness of the inverse operator A−1:X⟶Y implies boundedness, and hence, continuity, for the linear Schauder coordinate functionals,(18)X∋x=∑k=1∞ckxek↦cnx∈F,n∈ℕ,with(19)cn≤2A−1en,n∈ℕ,(see, e.g., [1–3]) as well as for the linear operators:(20)X∋x=∑k=1∞ckxek↦Snx≔∑k=1nckxek,Rnx≔∑k=n+1∞ckxek,n∈ℕ,with(21)I=Sn+Rn,n∈ℕ,(I is the identity operator on X) and(22)Sn≤A−1,Rn≤2A−1,n∈ℕ,(see, e.g., [3]).Remark 3.
Here and henceforth, we use the notation⋅ for the operator norm.


## 3. General Characterization

The following statement appears to be a perfect illustration of the profound observation by Schwartz found in [7] and chosen as the epigraph.Theorem 1 (general characterization of convergence).
LetX,⋅ be a (real or complex) Banach space with a Schauder basis enn∈ℕ and corresponding coordinate functionals cn⋅, n∈ℕ.
For a sequencexnn∈ℕ and a vector x in X,(23)xn⟶x,n⟶∞,iff(1)
∀k∈ℕ:ckxn⟶ckx, n⟶∞,(2)
∀ε>0∃K0∈ℕ∀K≥K0∀n∈ℕ:RKxn<ε.Proof.
“Only if” part.
Suppose that, for a sequencexnn∈ℕ and a vector x in X,(24)xn⟶x,n⟶∞.
Then, by the continuity of the Schauder coordinate functionalscn⋅, n∈ℕ, we infer that condition (1) holds.
Letε>0 be arbitrary. Then,(25)∃N∈ℕ∀n≥N:xn−x<ε4A−1.
Sincex∈X,(26)RKx≔∑k=K+1∞ckxek⟶0,K⟶∞,and hence,(27)∃K0∈ℕ∀K≥K0:RKx<ε2.
In view of (22), (25), and (27), we have(28)∀K≥K0,∀n≥N:RKxn=RKxn−RKx+RKx≤RKxn−x+RKx≤RKxn−x+RKx<2A−1ε4A−1+ε2=ε2+ε2=ε.
Furthermore, sincexn∈X, n=1,…,N−1, we can regard K0∈ℕ in (27) to be large enough so that(29)∀K≥K0,∀n=1,…,N−1:RKxn=∑k=K+1∞ckxnek<ε.
Thus, condition (2) holds as well.
This completes the proof of the “only if” part.
“If” part. Suppose that, for a sequence xnn∈ℕ and a vector x in X, conditions (1) and (2) are met.
For an arbitraryε>0 and K0∈ℕ, from condition (2), by condition (1),(30)∃N∈ℕ∀n≥N:SK0xn−x≤∑k=1K0ckxn−ckxek<ϵ3.
Sincex∈X, we can also regard that K0∈ℕ in condition (2) to be large enough so that(31)RK0x<ε3.
Then, in view of (21), (30), and (31) and by condition (2),(32)∀n≥N:xn−x=SK0xn−x+RK0xn−x≤SK0xn−x+RK0xn+RK0x<ε3+ε3+ε3=ε.
This concludes the proof of the “if” part and the entire statement.Remarks 4.
(i)
Condition (1) is the convergence of the coordinates ofxn to the corresponding coordinates of x relative to ekk∈ℕ.(ii)
Condition (2) signifies the uniform convergence of the Schauder expansions,(33)∑k=1∞ckxnek,of xn relative to ekk∈ℕ over n∈ℕ.Now, the combined characterization of convergence (Proposition3) is an instant corollary of the foregoing general characterization.


## 4. Characterization of Convergence in an Infinite-Dimensional Separable Hilbert Space

For an infinite-dimensional separable Hilbert spaceX,⋅,⋅,⋅ relative to an orthonormal basis enn∈ℕ, in view of (9), the general characterization of convergence (Theorem 1) acquires the following form.Corollary 1 (characterization of convergence in a separableHilbert space).
LetX,⋅,⋅,⋅ be a (real or complex) infinite-dimensional separable Hilbert space with an orthonormal basis enn∈ℕ.
For a sequencexnn∈ℕ and a vector x in X,(34)xn⟶x,n⟶∞,iff(1)
∀k∈ℕ:xn,ek⟶x,ek, n⟶∞,(2)
∀ε>0∃K∈ℕ∀n∈ℕ:∑k=K+1∞xn,ek2<ε.Remarks 5.
(i)
Condition (1) is the convergence of the Fourier coefficients ofxn to the corresponding Fourier coefficients of x relative to ekk∈ℕ.(ii)
Condition (2) signifies the uniform convergence of the Fourier series expansions,(35)∑k=1∞xn,ekek,of xn relative to ekk∈ℕ over n∈ℕ.(iii)
The characterization of convergence inlp (Proposition 1) for p=2 is now a particular case of the prior characterization.


## 5. Characterization of Convergence inc

Another immediate corollary of the general characterization of convergence (Theorem1) is the realization of the latter in the space c of convergent sequences equipped with ∞-norm (see (2)) relative to the standard Schauder basis enn∈ℤ+ (see Section 2).Indeed, inc relative to enn∈ℤ+, for an arbitrary x≔xkk∈ℕ,(36)x=∑k=0∞ckxek,withc0x=limm⟶∞xm,ckx=xk−c0x,k∈ℕ,(see (13)) and(37)SKx≔∑k=0Kckxek=limm⟶∞xm,x1−limm⟶∞xm,…,xK−limm⟶∞xm,0,…,RKx≔∑k=K+1∞ckxek=0,…,0︸K+1terms,xK+1−limm⟶∞xm,…,K∈ℤ+,(cf. (20)).Thus, the general characterization of convergence (Theorem1), in view of the obvious circumstance that, for any x≔xkk∈ℕ∈c, the sequence,(38)RKx=supk≥K+1xkn−limm⟶∞xmn,K∈ℤ+,is decreasing, acquires the following form.Corollary 2 (characterization of convergence inc).
In the (real or complex) spacec,(39)xknk∈ℕ≕xn⟶x≔xkk∈ℕ,n⟶∞,iff(1)
limm⟶∞xmn⟶limm⟶∞xm, n⟶∞, and ∀k∈ℕ:xkn⟶xk, n⟶∞,(2)
∀ε>0∃K∈ℤ+∀n∈ℕ:supk≥K+1xkn−limm⟶∞xmn<ε.Remarks 6.
(i)
Condition (1), beyond termwise convergence, includes convergence of the limits.(ii)
Condition (2) signifies the uniform convergence of the sequencesxknk∈ℕ to their respective limits over n∈ℕ.(iii)
The characterization of convergence inc0 (Proposition 2) is a mere restriction of the prior characterization to the subspace c0 of c.


## 6. Concluding Remark

As is easily seen, the general characterization of convergence (Theorem1) is consistent with the following characterization of compactness, which underlies the results of [8].Theorem 2 (characterization of compactness, Theorem III.7.4 in [3]).
In a (real or complex) Banach spaceX,⋅ with a Schauder basis, a set C is precompact (a closed set C is compact) iff(1)
C is bounded,(2)
∀ε>0∃K0∈ℕ∀K≥K0∀x∈C:RKx<ε.



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