Abstract

We provide an overview of linear spaces equipped with norms that may take on the value infinity but that otherwise satisfy the properties of ordinary norms. Spaces equipped with such norms fail to be topological vector spaces unless the norm is finite valued. We study the continuous linear transformations between such spaces, extending the theory of conventional linear analysis in often unanticipated ways.

Author information

Contact details are reproduced from the original publication and may be historical.

Gerald Beer

Dept. of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A.

gbeer@cslanet.calstatela.edu

Keywords

  • Extended norm
  • infinite valued norm
  • projection operator
  • projection complement
  • bornology
  • operator norm
  • effective domain
  • uniform convergence on bounded subsets
  • extended supremum norm
  • extended Lipschitz norm
  • distance basis

Mathematics Subject Classification

46B20, 46B28; 46A17, 46E15, 26A16

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Suggested citation

G. Beer. “Norms with Infinite Values.” Journal of Convex Analysis 22 (2015), No. 1, 37–60. https://doi.org/10.68381/jca22003

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.