In this paper we consider Bernstein's Lethargy Theorem (BLT) in the contextof Fr\'{e}chet spaces. Let $X$ be an infinite-dimensional Fr\'echet space andlet $\mathcal{V}=\{V_n\}$ 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 ofpositive 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 $$ \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n $$ forany $ n \geq n_o$. By using the above theorem, we prove both Shapiro's\cite{Sha} and Tyuremskikh's \cite{Tyu} theorems for Fr\'{e}chet spaces.Considering rapidly decreasing sequences, other versions of the BLT theorem inFr\'{e}chet spaces will be discussed. We also give a theorem improvingKonyagin's \cite{Kon} result for Banach spaces.
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机译：在本文中，我们在Fr'{e} chet空间的情况下考虑Bernstein的Lethargy定理（BLT）。假设$ X $是无穷维Fr'echet空间，而$ \ mathcal {V} = \ {V_n \} $是$ X $子空间的嵌套序列，使得$ \ bar {V_n} \ subseteq V_ { \ mathbb {N} $和$ X = \ bar {\ bigcup_ {n = 1} ^ {\ infty} V_n}中的任何$ n \ n + 1} $。$令$ e_n $为正数的递减序列趋于0。在$ \ sup \ {\ {dist}（x，V_n）\} $上的其他自然条件下，我们证明存在$ x \ in X $和$ n_o \ in \ mathbb {N} $这样$$ \ frac {e_n} {3} \ leq \ {dist}（x，V_n）\ leq 3 e_n $$ for any $ n \ geq n_o $。通过使用以上定理，我们证明了Fr \'{e} chet空间的Shapiro's \ cite {Sha}和Tyuremskikh's \ cite {Tyu}定理。考虑到快速递减的序列，BLT定理的其他版本inFr \'{e}将讨论chet空间。我们还给出了一个定理，用于改进Banach空间的Konyagin的\ cite {Kon}结果。
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