## Generalized Cauchy functional equation and characterizations of inner product spaces.(English)Zbl 0755.39007

A typical result is the following. Let $$X$$ be a real or complex linear vector space and $$F$$ be $$\mathbb{R}$$ or $$\mathbb{C}$$. Suppose that $$a_ j\in F\setminus\{0\}$$, $$b_ j\in F$$, $$c_ j\in F$$ ($$j=1,2,3,4$$) satisfy $$\sum_{j=1}^ n a_ j| b_ j|^ 2=\sum_{j=1}^ n a_ j| c_ j|^ 2=0$$, $$\sum_{j=1}^ n a_ j b_ j\bar c_ j=\sum_{j=1}^ n a_ j \bar b_ j c_ j=0$$, $$b_ j c_ k\neq b_ k c_ j$$ ($$j\neq k=1,2,\dots,n$$) and that $$\|\cdot\|: X\to F$$ satisfies $$\sum_{j=1}^ n a_ j\| b_ j x+c_ j y\|^ 2=0$$ ($$x,y\in X$$). Then $$X$$ is an inner product space.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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### References:

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