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Polarization inequality in complex normed spaces


We introduce a product in all complex normed vector spaces, which generalizes the inner product of complex inner product spaces. Naturally the question occurs whether the polarization inequality of line (3.1) is fulfilled. We show that the polarization inequality holds for the product from Definition 1.1. This also yields a new proof of the Cauchy-Schwarz inequality in complex inner product spaces, which does not rely on the linearity of the inner product. The proof depends only on the norm in the vector space.

Key words: Complex normed space, complex inner product space, Cauchy-Schwarz inequality.

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TitlePolarization inequality in complex normed spaces
SourceMethods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 301-308
MathSciNet   MR3707524
Milestones  Received 28/11/2016; Revised 24/04/2017
CopyrightThe Author(s) 2017 (CC BY-SA)

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Volker W. Thürey
Bremen, Germany

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Volker W. Thürey, Polarization inequality in complex normed spaces, Methods Funct. Anal. Topology 23 (2017), no. 3, 301-308.


@article {MFAT991,
    AUTHOR = {Thürey, Volker W.},
     TITLE = {Polarization inequality in complex normed spaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {23},
      YEAR = {2017},
    NUMBER = {3},
     PAGES = {301-308},
      ISSN = {1029-3531},
  MRNUMBER = {MR3707524},
       URL = {},


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