On maps which preserve equality of distance in F-spaces

TL;DR

This paper generalizes classical results by proving that any distance-equality preserving map between F-spaces that fixes zero must be linear, and characterizes linear isometries via their range properties.

Contribution

It extends the Mazur-Ulam theorem to F-spaces without requiring surjectivity, providing new insights into the structure of distance-preserving maps.

Findings

Maps preserving equality of distance with T(0)=0 are linear in F-spaces.

Linear isometries can be characterized by a simple property of their range.

Abstract

In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range.