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 $\bar{V_n} \subseteq V_{n+1}$ 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.