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Geometric Characterizations of Hilbert Spaces

  Published:2016-06-08
 Printed: Dec 2016
  • Francisco Javier García-Pacheco,
    Department of Mathematics, Universidad de Cádiz, Cadiz, Spain
  • Justin R. Hill,
    Temple College, Temple, Texas, USA
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Abstract

We study some geometric properties related to the set $\Pi_X:= \{ (x,x^* )\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^* (x )=1 \}$ obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $ (h,k )\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.
Keywords: Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summands Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summands
MSC Classifications: 46B20, 46C05 show english descriptions Geometry and structure of normed linear spaces
Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B20 - Geometry and structure of normed linear spaces
46C05 - Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
 

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