Subspace Condition for Bernstein Lethargy Theorem

Asuman GÜven Aksoy, Monairah Al-Ansari, Caleb Case, Qidi Peng

Published 2016-06-25Version 1

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 $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ 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 there exists $c \in (0,1]$ and an element $x_c \in X$ such that $$ c d_n \leq \rho(x_c, Y_n) \leq \min (4, \tilde{a}) c\, d_n. $$ Here, $\rho(x, Y_n)= \inf \{||x-y||: \,\,y\in Y_n\}$, $$\tilde{a} =\sup_{n\ge1}\sup_{\left \{q_{i} \right \}}\left \{a_{n_{i+1}-1}^{-3}\right \}$$ where the sequence $\{a_n\}$ is defined as: for all $ n \geq 1 $, $$ a_n = \inf_{l \geq n} \, \inf_{q \in \langle q_l, q_{l+1},\dots \rangle} \frac{\rho(q,Y_l)}{||q||} $$ in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$, and satisfies $\inf\limits_{n\ge1} a_n > 0$.