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.


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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