Bernstein's Lethargy Theorem in Fréchet spaces

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = ⋃∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there exists x ∈ X and no ∈ N such that en 3 ≤ dist(x, Vn) ≤ 3en for any n ≥ no. By using the above theorem, we prove both Shapiro’s [19] and Tyuremskikh’s [22] theorems for Fréchet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fréchet spaces will be discussed. We also give a theorem improving Konyagin’s [9] result for Banach spaces.