Geodesic
Geometric Characterizations of Hilbert Spaces
Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 769-775

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DOI

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.
DOI : 10.4153/CMB-2016-019-8
Mots-clés : 46B20, 46C05, Hilbert space, extreme point, smooth, L2-summands 
García-Pacheco, Francisco Javier; Hill, Justin R. Geometric Characterizations of Hilbert Spaces. Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 769-775. doi: 10.4153/CMB-2016-019-8
@article{10_4153_CMB_2016_019_8,
     author = {Garc{\'\i}a-Pacheco, Francisco Javier and Hill, Justin R.},
     title = {Geometric {Characterizations} of {Hilbert} {Spaces}},
     journal = {Canadian mathematical bulletin},
     pages = {769--775},
     year = {2016},
     volume = {59},
     number = {4},
     doi = {10.4153/CMB-2016-019-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-019-8/}
}
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