Orthogonalities, transitivity of norms and characterizations of Hilbert spaces

Horst Martini; Senlin Wu

doi:10.1216/RMJ-2015-45-1-287

2015 Orthogonalities, transitivity of norms and characterizations of Hilbert spaces

Horst Martini, Senlin Wu

Rocky Mountain J. Math. 45(1): 287-301 (2015). DOI: 10.1216/RMJ-2015-45-1-287

Abstract

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 $\sqrt2$, respectively) is a Hilbert space.

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Horst Martini and Senlin Wu "Orthogonalities, transitivity of norms and characterizations of Hilbert spaces," Rocky Mountain Journal of Mathematics 45(1), 287-301, (2015). https://doi.org/10.1216/RMJ-2015-45-1-287

Published: 2015

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