Bernstein's Lethargy Theorem in Frechet Spaces

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 $\overline{V_n}\subseteq V_{n+1}$ for any $n\in \mathbb{N}$ and $X=\overline{{\bigcup }_{n=1}^{\mathrm{\infty }}{V}_n}\mathrm{.}$ 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}$$suchthat$\frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n$ for any $n\ge n_o$. 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.