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Structural Properties of Extended Normed Spaces

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Abstract

We study some structural properties of real linear spaces equipped with norms that may take the value infinity but that otherwise satisfy the properties of conventional norms. A description is given of the finest locally convex topology weaker than the extended norm topology for which addition and scalar multiplication are jointly continuous. We also study bornologies and provide a characterization of relatively weakly compact sets in these spaces. It is shown that complemented and projection complemented closed subspaces can be different in extended Banach spaces. Particular attention is given to extended normed spaces whose subspace of vectors of finite norm has finite codimension.

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Authors and Affiliations

  1. Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA, 90032, USA

    G. Beer

  2. Department of Mathematics, La Sierra University, 4500 Riverwalk Parkway, Riverside, CA, 92515, USA

    J. Vanderwerff

Authors
  1. G. Beer
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  2. J. Vanderwerff
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Correspondence to J. Vanderwerff.

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à notre ami Lionel Thibault

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Beer, G., Vanderwerff, J. Structural Properties of Extended Normed Spaces. Set-Valued Var. Anal 23, 613–630 (2015). https://doi.org/10.1007/s11228-015-0331-x

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  • DOI: https://doi.org/10.1007/s11228-015-0331-x

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