On certain characterizations of inner product spaces (Q2839847)
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scientific article; zbMATH DE number 6187913
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On certain characterizations of inner product spaces |
scientific article; zbMATH DE number 6187913 |
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12 July 2013
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characterizations of inner product spaces
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norm inequality
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normed space
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inner product space
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On certain characterizations of inner product spaces (English)
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There are many characterizations of inner product spaces which are based on norm inequalities and various notions of orthogonality in normed spaces. The author shows that a normed linear space \(X\) is an inner product space if and only if, for each \(x, y \in X\) with \(\|x\| = \|y\| = 1\), there exists a real number \(c > 1\) such that \(\|cx + y\| = \|x + cy\|\) holds, or equivalently, there exists a real number \(t \in (0, 1/2)\) such that \(\|(1 - t)x + ty\| = \|tx + (1 - t)y\|\) holds; see [\textit{M. S. Moslehian} and \textit{J. M. Rassias}, Kochi J. Math. 6, 101--107 (2011; Zbl 1237.46008)].
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