Bernstein Lethargy Theorem and Reflexivity

TL;DR

This paper establishes a link between reflexivity in Banach spaces and a form of Bernstein's Lethargy Theorem, showing that certain nested subspace conditions imply the existence of elements with prescribed approximation properties.

Contribution

It proves the equivalence of reflexivity and a specific approximation property characterized by Bernstein's Lethargy Theorem in Banach spaces.

Findings

Reflexive Banach spaces satisfy a specific approximation property.

Nested subspace conditions lead to the existence of elements with prescribed distances.

The theorem characterizes reflexivity via approximation sequences.

Abstract

In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let $X$ be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence $\{d_n\}_{n\ge1}$ decrease to $0$. Suppose that $\{Y_n\}_{n\ge1}$ is a system of strictly nested subspaces of $X$ such that $\overline Y_n \subset Y_{n+1}$ for all $n\ge1$ and for each $n\ge1$, there exists $y_n\in Y_{n+1}\backslash Y_n$ such that the distance $\rho(y_n,Y_n)$ from $y_n$ to the subspace $Y_n$ satisfies $$ \rho(y_n,Y_n)=\|y_n\|. $$ Then, there exists an element $x\in X$ such that $\rho(x,Y_n)=d_n$ for all $n\ge1$.