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Isometric reflection vectors in Banach spaces. (English) Zbl 1081.46008

The notion of an isometric reflection vector is studied, meaning a vector \(u\neq0\) of a Banach space \(X\) such that for all elements \(v\) of some \(1\)-codimensional subspace the equation \(\| u+v\| =\| u-v\| \) holds. The main result is a new proof of a theorem due to J. Becerra Guerrero and A. Rodríguez Palacios [Rocky Mt.J. Math.30, No. 1, 63–83 (2000; Zbl 1001.46013)] which says that \(X\) is a Hilbert space if the set of isometric reflection vectors has nonempty interior.
In addition, there is a section dealing with parallelogram identity vectors; i.e., vectors for which the parallelogram identity holds true.

MSC:

46B04 Isometric theory of Banach spaces
46C15 Characterizations of Hilbert spaces

Citations:

Zbl 1001.46013
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Full Text: DOI

References:

[1] A. Aizpuru, F.J. García-Pacheco, \(L^2\); A. Aizpuru, F.J. García-Pacheco, \(L^2\)
[2] Amir, D., Characterizations of Inner Product Spaces (1986), Birkhäuser: Birkhäuser Basel · Zbl 0617.46030
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[4] Becerra, J.; Rodríguez, A., Isometric reflections on Banach spaces after a paper of Skorik and Zaidenberg, Rocky Mountain J. Math., 30, 63-83 (2000) · Zbl 1001.46013
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