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 $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ be an infinite sequence of numbers converging to $0$, and let $Y_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X$ be a sequence of closed nested subspaces in a Banach space $X$ with the property that $\overline{Y}_{n}\subset Y_{n+1}$ for all $n\ge1$. We prove that for any $c ın (0,1]$ there exists an element $x_c ın X$ such that$$ c d_n \leq \rho(x_c, Y_n) \leq \min (4, \tilde{a}) c\, d_n.$$Here, $\rho(x, Y_n)= ınf \{ ||x-y||: \,\,yın Y_n\}$, $$\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i} \right \}}\left \{ a_{n_{i+1}-1}^{-3}\right \}$$ where the sequence $\{a_n\}$ is defined as: for all $ n \geq 1 $,$$a_n = ınf_{l \geq n} \, ınf_{q ın \langle q_l, q_{l+1},\dots \rangle} \frac{\rho(q,Y_l)}{||q||}$$in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$, and satisfies $ınf?imits_{n\ge1} a_n > 0$. The sequence $\{n_i\}_{i\ge1}$ is given by$$n_1=1; n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} ?eq d_{n_{i}}\right \}, i\geq 1.$$