Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces

Document Type: Original Article

Authors

1 Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711, USA.

2 Institute of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA.

Abstract

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$
we can find $x_{c} \in X$, such that the distance $\rho(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies
$$
c d_n \leq \rho(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}.
$$
We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weakening his condition on the sequence $\{d_n\}$. The weakened condition on $d_n$ requires refinement of Borodin's construction to extract an element in $X$, whose distances from the nested subspaces are precisely the given values $d_n$.

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