Bernstein's Lethargy Theorem in Frechet Spaces

TL;DR

This paper extends Bernstein's Lethargy Theorem to Fréchet spaces, establishing existence results for elements with prescribed approximation properties and generalizing prior theorems in Banach and Fréchet spaces.

Contribution

It generalizes Bernstein's Lethargy Theorem to Fréchet spaces and proves related approximation theorems, including improvements of existing results.

Findings

Established BLT in Fréchet spaces under natural conditions.

Proved versions of Shapiro's and Tyuremskikh's theorems for Fréchet spaces.

Improved Konyagin's result for Banach spaces.

Abstract

In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Fr\'{e}chet spaces. Let $X$ be an infinite-dimensional Fr\'echet space and let $\mathcal{V}=\{V_n\}$ be a nested sequence of subspaces of $ X$ such that $ \bar{V_n} \subseteq V_{n+1}$ for any $ n \in \mathbb{N}$ and $ X=\bar{\bigcup_{n=1}^{\infty}V_n}.$ Let $ e_n$ be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on $\sup\{\{dist}(x, V_n)\}$, we prove that there exists $ x \in X$ and $ n_o \in \mathbb{N}$ such that $$ \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n $$ for any $ n \geq n_o$. By using the above theorem, we prove both Shapiro's \cite{Sha} and Tyuremskikh's \cite{Tyu} theorems for Fr\'{e}chet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fr\'{e}chet spaces will be discussed. We also give a theorem improving Konyagin's \cite{Kon} result for Banach spaces.