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

Aleksandr Danilovich Aleksandrov

Aleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...

 asked whether the existence of a single conservative distance for some mapping implies that it is an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

. Themistocles M. Rassias

Themistocles M. Rassias

Themistocles M. Rassias is a Greek mathematician, and a professor at the National Technical University of Athens , Greece. He has published more than 220 papers, 6 research books and 30 edited volumes in research Mathematics as well as 4 textbooks in Mathematics for university students...

posed the following problem: If X and Y are normed linear spaces and if T is a continuous and/ or surjective mapping which satisfies the so-called distance one preserving property (DOPP), is then T necessarily an isometry? This problem is called the Aleksandrov–Rassias problem. There have been several attempts in the mathematical literature by a number of researchers for the solution to this problem.