ORTHOGONALITIES, TRANSITIVITY OF NORMS AND CHARACTERIZATIONS OF HILBERT SPACES

We introduce three concepts, called I-vector, IP-vector, and P-vector, which are related to isosceles orthogonality and Pythagorean orthogonality in normed linear spaces. Having the Banach-Mazur rotation problem in mind, we prove that an almost transitive real Banach space, whose dimension is at lea...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2015-01, Vol.45 (1), p.287-301
Main Authors: MARTINI, HORST, WU, SENLIN
Format: Article
Language:English
Subjects:
46B20
46C05
46C15
52A21
Almost transitive norm
Banach-Mazur rotation problem
Birkhoff orthogonality
isometric reflection
isosceles orthogonality
James constant
Pythagorean orthogonality
Roberts orthogonality
Singer orthogonality
transitivity
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Summary:We introduce three concepts, called I-vector, IP-vector, and P-vector, which are related to isosceles orthogonality and Pythagorean orthogonality in normed linear spaces. Having the Banach-Mazur rotation problem in mind, we prove that an almost transitive real Banach space, whose dimension is at least three and which contains an I-vector (an IP-vector, a P-vector, or a unit vector whose pointwise James constant is √2, respectively) is a Hilbert space.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmj-2015-45-1-287