A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces

Abstract

We introduce fuzzy norm-preserving maps, which generalize the concept of fuzzy isometry. Based on the ideas from Vogt, 1973, and Väisälä, 2003, we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: let $X$, $Y$ be fuzzy normed spaces and let $f:X\to Y$ be a fuzzy norm-preserving surjection satisfying $f(0)=0$. Then $f$ is additive.