Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces

TL;DR

This paper improves a theorem by Bernstein on constructing elements in Banach spaces with prescribed deviations from nested subspaces, relaxing previous conditions and providing explicit bounds.

Contribution

It introduces a refined method to construct elements in Banach spaces with specified distances to nested subspaces, extending Borodin's and Bernstein's results.

Findings

Established existence of elements with controlled deviations in Banach spaces.

Weakened conditions on the deviation sequence for construction.

Provided explicit bounds for the deviations of constructed elements.

Abstract

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance $\rho(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \leq \rho(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weakening his condition on the sequence $\{d_n\}$. The weakened condition on $d_n$ requires refinement of Borodin's construction to extract an element in $X$, whose distances from the nested subspaces are precisely the given values $d_n$.