Characterization of inner product spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2014
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| _version_ | 1866914889914646528 |
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| author | Sain, Debmalya Paul, Kallol Debnath, Lokenath |
| author_facet | Sain, Debmalya Paul, Kallol Debnath, Lokenath |
| contents | 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 $(R^n,\|.\|_p).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_5016 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Characterization of inner product spaces Sain, Debmalya Paul, Kallol Debnath, Lokenath Functional Analysis Primary: 46B20, Secondary: 47A30 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 $(R^n,\|.\|_p).$ |
| title | Characterization of inner product spaces |
| topic | Functional Analysis Primary: 46B20, Secondary: 47A30 |
| url | https://arxiv.org/abs/1407.5016 |