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On orthogonally additive mappings. (English) Zbl 0569.39006
If (X,) is an orthogonality space and (Y,+) an Abelian group, then a mapping f:XY is said to be orthogonally additive if (1)f(x 1 +x 2 )=f(x 1 )+f(x 2 ) for all x 1 ,x 2 X with x 1 x 2 . Two of sixteen results obtained in this paper are as follows: Theorem 6. If (X,) is an orthogonality space, (Y,+) an Abelian group and g:XY an even solution of (1), then g is a quadratic mapping, i.e., g(x 1 +x 2 )+g(x 1 -x 2 )=2g(x 1 )+2g(x 2 ) for all x 1 ,x 2 X· Theorem 9. If (X,) is an inner product space and (Y,+) an Abelian group, then g:XY is an even solution of (1) if and only if there exists an additive mapping :RY such that g(x)=(x 2 ) for every xX.
Reviewer: H.Haruki

MSC:
39B52Functional equations for functions with more general domains and/or ranges
46C99Inner product spaces, Hilbert spaces
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