Geodesic
On isometric reflections in Banach spaces
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (1997) no. 1, pp. 212-247 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain the following characterization of Hilbert spaces. Let $E$ be a Banach space the unit sphere $S$ of which has a hyperplane of symmetry. Then $E$ is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group $\operatorname{Iso}E$ of $E$ has a dense orbit in $S'$ ; b) the identity component $G_0$ of the group $\operatorname{Iso}E$ endowed with the strong operator topology acts topologically irreducible on $E$. Some related results on infinite dimensional Coxeter groups generated by isometric reflections are given which allow us to analyse the structure of isometry groups containing sufficiently many reflections. 
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     author = {A. Skorik and M. G. Zaidenberg},
     title = {On isometric reflections in {Banach} spaces},
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     year = {1997},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_1997_4_1_a12/}
}
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A. Skorik; M. G. Zaidenberg. On isometric reflections in Banach spaces. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (1997) no. 1, pp. 212-247. http://geodesic.mathdoc.fr/item/JMAG_1997_4_1_a12/