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A conditional Cauchy equation on normed spaces. (English) Zbl 0807.39010
The main theorem of the paper says that if a function f from a real normed linear space (X,·) of dimension 3 into an Abelian group (Y,+) satisfies the conditional Cauchy equation f(x+y)=f(x)+f(y), whenever x=y, then f is additive. This result generalizes some theorems of C. Alsina and J. L. Garcia-Roig [On a conditional Cauchy equation on rhombuses (to appear)] and its proof contains an interesting connectivity theorem. An application to a class of orthogonal additive mappings is given.

MSC:
39B52Functional equations for functions with more general domains and/or ranges