The Aleksandrov problem in linear 2-normed spaces

Abstract

We introduce the concept of 2-isometry which is suitable to represent the notion of area preserving mappings in linear 2-normed spaces. And then we obtain some results for the Aleksandrov problem in linear 2-normed spaces.

On the generalized 2-isometry

“…By [2], we obtain that r = k, that is f (rx) = rf (x) for all r ∈ R + . Furthermore, for all r ∈ R, we have f (rx) = rf (x).…”

“…Other generalizations of the Mazur-Ulam theorem can be seen in [1,3,6,[9][10][11][12]14]. Recently, Chu [2] introduces the concept of 2-isometry which is suitable to represent the notion of area preserving mapping in 2-normed linear spaces and obtain that a 2-isometry f is affine if f preserves the 2-colinearity (i.e. x, y, z are colinear implies f (x), f (y), f (z) are colinear).…”

ON STRONG AND WEAK CONVERGENCE IN $n-$HILBERT SPACES

“…The corresponding theory of n-inner product spaces was then established by Misiak ([10]). Since then, various aspects of the theory have been studied, for instance the study of Mazur-Ulam theorem and Aleksandrov problem in n-normed spaces are done in [1,2], the study of operators in n-Banach space is done in [5,11], and many others.…”

THE ALEKSANDROV PROBLEM AND THE MAZUR-ULAM THEOREM ON LINEAR n-NORMED SPACES

“…Since 2004, the Aleksandrov problem has been discussed in the n-normed spaces (n ≥ 2) (see [2,3,4,5]). Chu, Lee and Park proved:…”