Subspace condition for Bernstein's lethargy theorem

Abstract

In this paper, we consider a condition on subspaces in order to improve bounds given in Bernstein s lethargy theorem for Banach spaces. Let d1 ≥ d2 ≥ . . . dn ≥ · · · > 0 be an infinite sequence of numbers converging to 0, and let Y1 ⊂ Y2 ⊂ · · · ⊂ Yn ⊂ · · · ⊂ X be a sequence of closed nested subspaces in a Banach space X with the property that Y n ⊂ Yn+1 for all n ≥ 1. We prove that for any c ∈ (0, 1] there exists an element xc ∈ X such that cdn ≤ ρ(xc, Yn) ≤ min(4, a )c dn. Here, ρ(x, Yn) = inf{||x − y|| : y ∈ Yn}, a = sup i≥1 sup {qi} { a −3 ni+1−1 } where the sequence {an} is defined as: for all n ≥ 1, an = inf l≥n inf q∈⟨ql ,ql+1,... ⟩ ρ(q, Yl) ||q|| in which each point qn is taken from Yn+1 Yn , and satisfies inf n≥1 an > 0. The sequence {ni}i≥1 is given by n1 = 1; ni+1 = min { n ≥ 1 : dn a 2 n ≤ dni } , i ≥ 1