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

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 $$ c d_n \leq ρ(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ 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 $d_n$.

18 pages

References in corpus (1)

• On Shapiro's lethargy theorem and some applications

Cited by in corpus (1)

• Subspace Condition for Bernstein's Lethargy Theorem