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.[1] They proved the Mazur–Ulam theorem stating that every 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 distance that is preserved by a mapping implies that it is an isometry, as it does for Euclidean spaces by the Beckman–Quarles theorem. 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 such that whenever vectors x and y in X satisfy , then (the distance one preserving property or DOPP), is T then necessarily an isometry?[2]

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

References

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  1. ^ S. Mazur and S. Ulam, Sur les transformationes isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194(1932), 946–948.
  2. ^ Tan, Liyun; Xiang, Shuhuang (January 2007). "On the Aleksandrov–Rassias problem and the Hyers–Ulam–Rassias stability problem". Banach Journal of Mathematical Analysis. 1 (1): 11–22. doi:10.15352/bjma/1240321551.