Bernstein Lethargy Theorem and Reflexivity

2018/03/27 by Asuman Güven Aksoy, Aksoy, Asuman Güven, Qidi Peng +1
Mathematics · Computer Science · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1803.09874

Abstract

In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let X be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence \d_n\_n≥1 decrease to 0. Suppose that \Y_n\_n≥1 is a system of strictly nested subspaces of X such that Y_n ⊂ Y_n+1 for all n≥1 and for each n≥1, there exists y_n∈ Y_n+1\backslash Y_n such that the distance ρ(y_n,Y_n) from y_n to the subspace Y_n satisfies ρ(y_n,Y_n)=‖y_n‖. Then, there exists an element x∈ X such that ρ(x,Y_n)=d_n for all n≥1.

Related

• Subspace Condition for Bernstein's Lethargy Theorem
• Bernstein's Lethargy Theorem in Frechet Spaces
• Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces
• Extensions of Bernstein's Lethargy Theorem
• On a theorem of Nikol'skii