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.


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


Zbl 1001.46013
Full Text: DOI


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