A characterisation of Hilbert spaces via orthogonality and proximinality
2005 ◽
Vol 71
(1)
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pp. 107-111
Keyword(s):
In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.
1979 ◽
Vol 31
(3)
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pp. 628-636
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Keyword(s):
2002 ◽
Vol 133
(3)
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pp. 515-530
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1999 ◽
Vol 59
(2)
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pp. 177-180
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Keyword(s):
1988 ◽
Vol 37
(1)
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pp. 149-160
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2010 ◽
Vol 10
(2)
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pp. 325-348
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1977 ◽
Vol 29
(5)
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pp. 963-970
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Keyword(s):
2012 ◽
Vol 20
(1)
◽
pp. 329-344