Bernstein's Lethargy Theorem in Frechet Spaces

2015/03/20 by Asuman Güven Aksoy, Aksoy, Asuman Guven, Grzegorz Lewicki +1 · 1 citation
Computer Science · Mathematics · #41A25 #41A50 #41A65 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1503.06190

openalex publication_date 2015/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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=\V_n\ be a nested sequence of subspaces of X such that V_n ⊆ V_n+1 for any n ∈ ℕ and X=\bigcup_n=1^∞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 ∈ X and n_o ∈ ℕ such that (e_n)/(3) ≤ \dist(x,V_n) ≤ 3 e_n for any n ≥ n_o. By using the above theorem, we prove both Shapiro's \citeSha and Tyuremskikh's \citeTyu 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 \citeKon result for Banach spaces.

Citations

• On Shapiro's lethargy theorem and some applications

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• On a theorem of Nikol'skii

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