Approximating points of a Banach space by points of an operator image

Анотація

Answering one problem that has its origins in quantum mechanics, we prove that for any sequence ${\left(A_n\right)}_{n\in \mathrm{\mathbb{N}}}$ of convex nowhere dense sets in a Banach space $X$ and any sequence ${\left({\epsilon }_n\right)}_{n=1}^{\infty }$ of positive real numbers with $\underset{n\to \infty }{lim}{\epsilon }_n=0$, the set $A=\left\{x\in X:\forall n\in \mathrm{\mathbb{N}} \exists a\in A_n ||x-a||<{\epsilon }_n\right\}$ is nowhere dense in $X$.