Characterization of inner product spaces. (English) Zbl 1420.46025
Summary: We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterizes a real inner product space of dimension \(>2\). We conjecture that a finite dimensional real smooth normed space of dimension \(>2\) is an inner product space iff given any element on the unit sphere there exists a strongly orthonormal Hamel basis in the sense of Birkhoff-James containing that element. This is substantiated by our result on the spaces \((\mathbb{R}^n,\Vert \cdot\Vert_p)\).
MSC:
| 46C15 | Characterizations of Hilbert spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 41A50 | Best approximation, Chebyshev systems |
Keywords:
strong orthogonality; best approximation; best coapproximation; strictly convex space; inner product spaceReferences:
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