The theory of isometries in the framework of Banach spaces has its beginning in a paper by StanisĆaw Mazur and StanisĆaw M. Ulam in 1932. [1] They proved the MazurâUlam 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 BeckmanâQuarles theorem. Themistocles M. Rassias posed the following problem:
AleksandrovâRassias Problem. If X and Y are normed linear spaces and if T : X â Y 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.
The theory of isometries in the framework of Banach spaces has its beginning in a paper by StanisĆaw Mazur and StanisĆaw M. Ulam in 1932. [1] They proved the MazurâUlam 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 BeckmanâQuarles theorem. Themistocles M. Rassias posed the following problem:
AleksandrovâRassias Problem. If X and Y are normed linear spaces and if T : X â Y 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.