On the mappings preserving equality of 2-distance

Abstract

In this paper, we consider 2-isometries, 2-continuous mappings and mappings preserving equality of 2-distance and investigate the relation between the three mappings in linear 2-normed spaces. We prove that if f : E → F is a mapping preserving equality of 2-distance with its gauge function injective, f is 2-continuous and the range of f contains a segment, then f is affine. Moreover, when dimensionE > 2, the condition that f is 2-continuous could be eliminated.

Keywords: Linear 2-normed space; Mapping preserving equality of 2-distance; 2-Isometry; 2-Continuous function.

Quaestiones Mathematicae 33(2010), 11–20.