On a characterization of convergence in Banach spaces with a Schauder basis

We extend the well-known characterizations of convergence in the spaces $l_p$ ($1\le p<\infty$) of $p$-summable sequence and $c_0$ 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.


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., [4]). Thus, the natural question: 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 spaces l p (1 ≤ p < ∞) of p-summable sequences with p-norm only one additional condition is needed. The following characterizations of convergence in the foregoing spaces are well-known.
• Condition (2) signifies the uniform convergence of the series ∞ k=1 x (n) k p to their respective sums over n ∈ N.
• Condition (2) signifies the uniform convergence of the sequences x (n) k k∈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: where · stands for p-norm · p (1 ≤ p < ∞) or ∞-norm, respectively, and the mapping R K : X → X, K ∈ N, (X := l p (1 ≤ p < ∞) or X := c 0 ) is defined as follows: Thus, we have the following combined characterization encompassing both l p (1 ≤ p < ∞) and c 0 .
In view of the fact that both l p (1 ≤ p < ∞) and c 0 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 an infinite-dimensional separable Hilbert space and the Banach space c of convergent sequences.

Preliminaries
Here, we briefly outline certain preliminaries essential for our discourse.
The set of F-termed sequences c k e k converges in X with termwise linear operations and the norm c k e k is a Banach space and the linear operator is subject to the Inverse Mapping Theorem (see, e.g., [1, [3][4][5]). The boundedness of the inverse operator A −1 : X → Y implies boundedness, and hence, continuity, for the linear Schauder coordinate functionals (see, e.g., [3][4][5]) as well as for the linear operators (I is the identity operator on X) and (2.5) S n ≤ A −1 and R n ≤ 2 A −1 , n ∈ N, (see, e.g., [3]).
Remark 2.1. Here and henceforth, we use the notation · for the operator norm.

General Characterization
The following statement appears to be a perfect illustration of the profound observation by Jacob T. Schwartz found in [7] and chosen as the epigraph.

Theorem 3.1 (General Characterization of Convergence).
Let (X, · ) be a (real or complex) Banach space with a Schauder basis {e n } n∈N and corresponding coordinate functionals c n (·), n ∈ N.
For a sequence (x n ) n∈N and a vector x in X, Proof.
"Only if " part. Suppose that, for a sequence (x n ) n∈N and a vector x in X, Then, by the continuity of the Schauder coordinate functionals c n (·), n ∈ N, we infer that condition (1) holds.
This completes the proof of the "only if " part.
"If " part. Suppose that, for a sequence (x n ) n∈N and a vector x in X, conditions (1) and (2) are met.
For an arbitrary ε > 0 and K 0 ∈ N from condition (2), by condition (1), Since x ∈ X, we can also regard that K 0 ∈ N in condition (2) to be large enough so that Then, in veiw of (2.4), (3.3), and (3.4) and by condition (2), This concludes the proof of the "if " part and the entire statement.
• Condition (1) is the convergence of the coordinates of x n to the corresponding coordinates of x relative to {e k } k∈N .
• Condition (2) signifies the uniform convergence of the Schauder expansions

Characterization of Convergence in an Infinite-Dimensional Separable Hilbert Space
For an infinite-dimensional separable Hilbert space (X, (·, ·), · ) relative to an orthonormal basis {e n } n∈N , in view of (2.1), the General Characterization of Convergence (Theorem 3.1) acquires the following form. Let (X, (·, ·), · ) be a (real or complex) infinite-dimensional separable Hilbert space with an orthonormal basis {e n } n∈N .
• Condition (1) is the convergence of the Fourier coefficients of x n to the corresponding Fourier coefficients of x relative to {e k } k∈N .
• Condition (2) signifies the uniform convergence of the Fourier series expansions ∞ k=1 (x n , e k )e k of x n relative to {e k } k∈N over n ∈ N.
• The Characterization of Convergence in l p (Proposition 1.1) for p = 2 is now a particular case of the prior characterization.

Characterization of Convergence in c
Another immediate corollary of the General Characterization of Convergence (Theorem 3.1) is the realization of the latter in the space c of convergent sequences equipped with ∞-norm (see (1.1)) relative to the standard Schauder basis {e n } n∈Z+ (see Preliminaries).
Indeed, in c relative to {e n } n∈Z+ , for an arbitrary x := (x k ) k∈N , (see (2.2)) and Thus, the General Characterization of Convergence (Theorem 3.1), in view of the obvious circumstance that, for any x := (x k ) k∈N ∈ c, the sequence is decreasing, acquires the following form. k → x k , n → ∞; (2) ∀ ε > 0 ∃ K ∈ Z + ∀ n ∈ N : sup Remarks 5.1.
• Condition (2) signifies the uniform convergence of the sequnces x (n) k k∈N to their respective limits over n ∈ N.
• The Characterization of Convergence in c 0 (Proposition 1.2) is a mere restriction of the prior characterization to the subspace c 0 of c.

Concluding Remark
As is easily seen, the General Characterization of Convergence (Theorem 3.1) is consistent with the following characterization of compactness, which underlies the results of [6]. In a (real or complex) Banach space (X, · ) with a Schauder basis, a set C is precompact (a closed set C is compact) iff