Decompositions in direct sum of seminormed vector spaces and Mazur-Ulam theorem

TL;DR

This paper generalizes the Mazur-Ulam theorem to seminormed vector spaces, establishing conditions for when distance-preserving maps are linear, extending classical results in the geometry of normed spaces.

Contribution

It provides necessary and sufficient conditions for isometries to be linear in seminormed spaces, broadening the scope of the classical Mazur-Ulam theorem.

Findings

Generalization of Mazur-Ulam theorem to seminormed spaces

Identification of conditions for linearity of isometries

Extension of isometric mapping theory in vector spaces

Abstract

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur--Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.