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Isosceles orthogonally additive mappings and inner product spaces. (English) Zbl 0865.46012
Summary: In a normed vector space (X,|·|), consider James’ isosceles orthogonality, i.e., xy|x+y|=|x-y|. It is known that any odd, orthogonally additive mapping from X into an Abelian group is unconditionally additive whenever dimX3. In this paper a complementary result is presented: the existence of a nontrivial even orthogonally additive mapping characterizes inner product spaces for dimX2. The proof uses some interesting connectivity theorems.
MSC:
46C15Characterizations of Hilbert spaces
46C05Hilbert and pre-Hilbert spaces: geometry and topology