Aleksandrov–Rassias problem
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. 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 : X → Y is a continuous and/or surjective mapping which satisfies the so-called distance one preserving property, 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.
