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# Geometric Characterizations of Hilbert Spaces

Published:2016-06-08
Printed: Dec 2016
• Francisco Javier García-Pacheco,
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