On the Scottish Book Problem 155 by Mazur and Sternbach

TL;DR

This paper investigates a problem in Banach space theory, proving that local isometries are global under separability or surjectivity assumptions, advancing understanding of isometric mappings.

Contribution

It establishes that local isometries between Banach spaces are globally isometric if the domain is separable or the map is surjective, extending previous results.

Findings

Local isometries are global under separability.

Surjective local isometries are global.

Advances understanding of isometric mappings in Banach spaces.

Abstract

Problem 155 of the Scottish Book asks whether every bijection $U\colon X\to Y$ between two Banach spaces $X, Y$ with the property that, each point of $X$ has a neighborhood on which $U$ is isometric, is globally isometric on $X$. We prove that this is true under the additional assumption that $X$ is separable and the weaker assumption of surjectivity instead of bijectivity.