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Geometric Characterizations of Hilbert Spaces

Published online by Cambridge University Press:  20 November 2018

Francisco Javier García-Pacheco  and
Justin R. Hill
Francisco Javier García-Pacheco
Affiliation:
Department of Mathematics, Universidad de Cádiz, Cadiz, Spain e-mail: [email protected]
Justin R. Hill
Affiliation:
Temple College, Temple, Texas, USA e-mail: [email protected]
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Abstract

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We study some geometric properties related to the set

$${{\Pi }_{X}}\,:=\left\{ \left( x,\,{{x}^{*}} \right)\,\in \,{{\text{S}}_{X}}\,\times \,{{\text{S}}_{{{X}^{*}}}}\,:\,{{x}^{*}}\left( x \right)\,=\,1 \right\}$$

obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $\left( h,\,k \right)\,\in \,H\,{{\oplus }_{2}}\,H$ to ${{\Pi }_{H}}$ for $H$ a Hilbert space.

Keywords

46B2046C05Hilbert spaceextreme pointsmoothL2-summands
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 769 - 775
Copyright
Copyright © Canadian Mathematical Society 2016

References

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