Aleksandrov–Rassias problem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932.[1] They proved that each isometry of a normed real linear space onto a normed real linear space is a linear mapping up to translation. In 1970, Aleksandr Danilovich Aleksandrov asked whether the existence of a single conservative distance for some mapping implies that it is an isometry. Themistocles M. Rassias posed the following problem:

Aleksandrov–Rassias Problem. If X and Y are normed linear spaces and if T : XY is a continuous and/or surjective mapping which satisfies the so-called distance one preserving property (DOPP), is then T necessarily an isometry?

There have been several attempts in the mathematical literature by a number of researchers for the solution to this problem.

References[edit]

  1. ^ S. Mazur and S. Ulam, Sur les transformationes isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194(1932), 946–948.