Full length articleBernstein'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. Let en be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup{dist(x,Vn)}, we prove that there exist x∈X and no∈N such thaten3≤dist(x,Vn)≤3en for any n≥no. By using the above theorem, as a corollary we obtain classical Shapiro's (1964) and Tyuriemskih's (1967) theorems for Banach spaces. Also we prove versions of both Shapiro's (1964) and Tyuriemskih's (1967) 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 (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Fréchet spaces, such as Orlicz spaces generated by s-convex functions and locally bounded Fréchet spaces.