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

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