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Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki
Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki
CMC Faculty Publications and Research
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 = S∞ 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, V …