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

## TL;DR

This paper proves that in Banach spaces, the set of points approximable by sequences from convex nowhere dense sets with vanishing error is itself nowhere dense, addressing a problem rooted in quantum mechanics.

## Contribution

It establishes a new result about the structure of approximable points in Banach spaces, linking convex nowhere dense sets and their approximation properties.

## Key findings

- The set of points approximable by convex nowhere dense sets with decreasing error is nowhere dense.
- The result applies to sequences of convex nowhere dense sets in Banach spaces.
- Addresses a problem originating from quantum mechanics.

## Abstract

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