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

TL;DR

This paper generalizes the understanding of convergence in Banach spaces with a Schauder basis, extending known results from classical sequence spaces to a broader class of Banach spaces.

Contribution

It provides a new characterization of convergence in Banach spaces with a Schauder basis, unifying and extending previous results for classical sequence spaces.

Findings

Characterization of convergence in Banach spaces with a Schauder basis.

Corollaries for convergence in separable Hilbert spaces.

Corollaries for the space of convergent sequences.

Abstract

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.