On a Theorem of S. N. Bernstein for Banach Spaces

Asuman G. Aksoy, Qidi Peng

Published 2016-05-15Version 1

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $\{Y_n\}$ is a nested sequence of subspaces of $ X$ such that $ Y_n \subset Y_{n+1}$ and $\overline{Y_n} \subset Y_{n+1}$ for any $ n \in \mathbb{N} $, and if $\{d_n \}$ be a decreasing sequence of positive numbers tending to 0, then for any $0< c\leq 1$ there exists $ x_c \in X$ such that the distance $\rho(x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$ c d_n \leq \rho(x_c,Y_n) \leq 4c d_n. $$ We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $\{d_n\}$. Lastly, we compare subsequences $\{d_{\varphi(n)}\}$ under different choices of $\varphi(n)$ and examine their effects on approximation.