On the Aleksandrov-Rassias Problems on Linearn-Normed Spaces

Yumei Ma

doi:10.1155/2013/394216

Abstract

This paper generalizes T. M. Rassias' results in 1993 ton-normed spaces. IfXandYare two realn-normed spaces andYisn-strictly convex, a surjective mappingf:X→Ypreserving unit distance in both directions and preserving any integer distance is ann-isometry.

Highlights

A mapping f : X → Y is called an isometry if f satisfies dY(f(x), f(y)) = dX(x, y) for all x, y ∈ X, where dX(⋅, ⋅) and dY(⋅, ⋅) denote the metrics in the spaces X and Y, respectively
Examine whether the existence of a single conservative distance for some mapping T implies that T is an isometry
This question is of great significance for the Mazur-Ulam Theorem [2]

Summary

Introduction

If X and Y are two real n-normed spaces and Y is n-strictly convex, a surjective mapping f : X → Y preserving unit distance in both directions and preserving any integer distance is an n-isometry. A mapping f : X → Y is called an isometry if f satisfies dY(f(x), f(y)) = dX(x, y) for all x, y ∈ X, where dX(⋅, ⋅) and dY(⋅, ⋅) denote the metrics in the spaces X and Y, respectively. Suppose that f : X → Y is a surjective mapping that satisfies SDOPP. Let X and Y be two real normed linear spaces such that one of them has a dimension greater than one.

Results

Conclusion