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 d(1) >= d(2) >= ... d(n) >= ... > 0 be an infinite sequence of numbers converging to 0, and let Y-1 subset of Y-2 subset of ... subset of Y-n subset of ... subset of X be a sequence of closed nested subspaces in a Banach space X with the property that (Y) over bar (n) subset of Yn+1 for all n >= 1. We prove that for any c is an element of (0; 1] there exists an element x(c) is an element of X such that
cd(n) <= rho (x(c); Y-n) <= min(4; (a) over tilde) cd(n).
Here, rho(x; Y-n) = inf {parallel to x - y parallel to : y is an element of Y-n}
(a) over tilde = sup sup (i >= 1{qi}) {a(ni+1)(-3)-1}
where the sequence {a(n)} is defined as: for all n >= 1,
a(n) = inf(l >= n) inf (q is an element of < ql; ql + 1 ,...>) rho(q; Y-l)/parallel to q parallel to
in which each point q(n) is taken from Yn+1\Y-n, and satisfies inf(n >= 1) a(n) > 0. The sequence {n(i)}(i >= 1) is given by
n(1) = 1; n(i+1) = min {n >= 1 : d(n)/a(n)(2) <= d(ni)}; i >= 1.