Bernstein Lethargy Theorem and Reflexivity

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 $n\ge1$.