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 $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ 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 $c \in (0,1]$, there exists 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, $\rho(x, Y_n)= \inf \{ ||x-y||: \,\,y\in Y_n\}$, $$\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i} \right \}}\left \{ a_{n_{i+1}-1}^{-3}\right \}$$ where the sequence $\{a_n\}$ is defined as: for all $n \geq 1$, $$a_n = \inf_{l \geq n} \, \inf_{q \in \langle q_l, q_{l+1},\dots \rangle} \frac{\rho(q,Y_l)}{||q||}$$ in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$, and satisfies $\inf\limits_{n\ge1} a_n > 0$. The sequence $\{n_i\}_{i\ge1}$ is given by %Theorem \ref{100}, $\{n_i\}$ satisfying (\ref{ni}) and $n_{i}\leq n<n_{i+1}$. $$n_1=1;~n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} \leq d_{n_{i}}\right \},~i\geq 1.$$