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Let X and Y be normed linear spaces. A mapping T : X ! Y is called preserving the distance r if for all x,y of X with kx ykX = r then kT(x) T(y)k = r. In this paper, we provide an overall account of the development of the Aleksandrov problem, the Aleksandrov-Rassias problem for mappings which preserve distances and details for the Hyers-Ulam-Rassias stability problem.

On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem