The Aleksandrov-Rassias Problem In Inner Product Spaces

Let T:X→ Y be a mapping between two metric spaces X and Y. The Aleksan-drov problem asks whether such a mapping preserving a single distance is an isometry.The Aleksandrov-Rassias problem asks whether such a mapping, which preserves the two distances with a nonintegral ratio, is an isometry.In this paper, we get some results for Aleksandrov-Rassias problem in inner product spaces by introducing a concept of semi-parallelogram and generalizing the parallelo-gram law. Details are as follows.In §3.1, we introduce a concept of semi-parallelogram and study its property.In §3.2, we obtain some results for the Aleksandrov-Rassias problem concerning the semi-parallelogram and prove the following theorem. Let X and Y be real inner product spaces with dimX ≥ 2 and T : X →Y a mapping. Suppose that T preserves 1 and k((?)+(?)) for some positive integers k,l, and m. Then T is an affine isometry.In §3.3, we generalize the parallelogram law.In §3.4, we obtain an important result for the Aleksandrov-Rassias problem by using the generalized parallelogram law as follows. Let X and Y be real inner product spaces with dimX≥n(n≥1) and T : X →Y a mapping. Suppose that T preserves 1 and k((?)+(?)) for some positive integers k,l, and m not all 1. Then T is an affine isometry.Moreover, we study the relationships between several norms in linear spaces which plays a fundamental role in the Aleksandrov problem. Details are as follows.In §4.1, we prove the quasi-convex-norm and the common norm are equivalent.In §4.2, we discuss the relationship between the 2-norm and the quasi-convex-2-norm and obtain a useful result.