This is an announcement for the paper “On a Theorem of S. N. Bernstein for Banach Spaces” by Asuman G. Aksoy and Qidi Peng.
Abstract: This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $(Y_n)$ is a nested sequence of subspaces of $X$ such that $Y_n\subset Y_{n+1}$ and $\bar{Y_n}\subset Y_{n+1}$ for any $n\in\mathbb{N}$ and if $(d_n)$ be a decreasing sequence of positive numbers tending to 0, then for any $0<c\leq 1$ there exists $x_c\in X$ such that the distance $\rho (x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$cd_n\leq\rho(x_c, Y_n)\leq 4c d_n$$. We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $(d_n)$. Lastly, we compare subsequences $(d(\phi_n))$ under different choices of $(\phi_n)$ and examine their effects on approximation.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04592