Subspace Condition for Bernstein's Lethargy Theorem

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 $Y_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X$beasequenceofclosednestedsubspacesinaBanachspace$X$withthepropertythat$\overline{Y}{n}\subset Y{n+1}$forall$n\ge1$. We prove that for any $c\in (0,1]$, there exists an element $x_c\in X$ such that $$c{d}_n\le \rho (x_c,Y_n)\le \min (4,\tilde{a})c{d}_n\mathrm{.}$$ Here, $\rho (x,Y_n)=\inf \{\mathrm{\mid }\mathrm{\mid }x-y\mathrm{\mid }\mathrm{\mid }:y\in Y_n\}$, $$\tilde{a}=\underset{i\ge 1}{\sup }\underset{\left\{q_i\right\}}{\sup }\left\{a_{n_{i+1}-1}^{-3}\right\}$$ where the sequence $\{a_n\}$ is defined as: for all $n\ge 1$, $$a_n=\underset{l\ge n}{\inf }\underset{q\in ⟨q_l,q_{l+1},\dots ⟩}{\inf }\frac{\rho (q,Y_l)}{\mathrm{\mid }\mathrm{\mid }q\mathrm{\mid }\mathrm{\mid }}$$ in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$,andsatisfies$\inf\limits_{n\ge1} a_n > 0$\mathrm{.}Thesequence${n_i}_{i\ge1}$isgivenby${n_i}$ satisfying (\ref{ni}) and $n_i\le n<n_{i+1}$. $$n_1=1;n_{i+1}=\min \{n\ge 1:\frac{d_n}{a_n^2}\le d_{n_i}\},i\ge 1.$$