Subspace Condition for Bernstein's Lethargy Theorem

Abstract

In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1\ge d_2\ge \dots d_n\ge \cdots >0$ be an infinite sequence of numbers converging to $0$, and let $Y_1\subset Y_2\subset \cdots \subset Y_n\subset \cdots \subset X$ 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 $n\ge 1$. We prove that for any $c\in (0,1]$, there exists an element $x_c\in X$ such that $$c{d}_n\le \rho (x_c,Y_n)\le \min (4,\tilde{a})c{d}_n.$$ Here, $\rho (x,Y_n)=\inf \{||x-y||:y\in Y_n\}$, $$\tilde{a}=\underset{i\ge 1}{\sup }\underset{\left\{q_i\right\}}{\sup }\left\{a_{n_{i+1}-1}^{-3}\right\}$$ where the sequence $\{a_n\}$ is defined as: for all $n\ge 1$, $$a_n=\underset{l\ge n}{\inf }\underset{q\in ⟨q_l,q_{l+1},\dots ⟩}{\inf }\frac{\rho (q,Y_l)}{||q||}$$ in which each point $q_n$ is taken from $Y_{n+1}\setminus Y_n$, and satisfies $\underset{n\ge 1}{\inf }{a}_n>0$. The sequence $\{n_i{\}}_{i\ge 1}$ is given by %Theorem \ref{100}, $\{n_i\}$ satisfying (\ref{ni}) and $n_i\le n<n_{i+1}$. $$n_1=1;n_{i+1}=\min \left\{n\ge 1:\frac{d_n}{a_n^2}\le d_{n_i}\right\},i\ge 1.$$

9 pages

References in corpus (2)

• On Shapiro's lethargy theorem and some applications
• Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces