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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 $(X, \Vert \cdot \Vert)$ of dimension $\ge 3$ into an Abelian group $(Y,+)$ satisfies the conditional Cauchy equation $f(x + y) = f(x) + f(y)$, whenever $\Vert x \Vert = \Vert y \Vert$, then $f$ is additive. This result generalizes some theorems of {\it C. Alsina} and {\it J. L. Garcia}-{\it 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.

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