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Duality

In: Convergence Structures and Applications to Functional Analysis

Author

Listed:
  • R. Beattie

    (Mount Allison University, Department of Mathematics and Computer Science)

  • H.-P. Butzmann

    (UniversitΓ€t Mannheim, FakultΓ€t fΓΌr Mathematik und Informatik)

Abstract

If E is a Hausdorff locally convex topological vector space, then there is no vector space topology on 𝓛E making the evaluation Ο‰ e : 𝓛E x E β†’ 𝕂 continuous unless E is a normed space. This is a very serious shortcoming and was one of the main motivations for the study of convergence structures. Clearly the continuous convergence structure on 𝓛E makes evaluation continuous for every convergence vector space E. The resulting space 𝓛 c E is called the dual space of E. We sometimes also call it the continuous dual or c-dual of E in order to distinguish it from the strong dual of a locally convex topological vector space or the normed dual of a normed space.

Suggested Citation

  • R. Beattie & H.-P. Butzmann, 2002. "Duality," Springer Books, in: Convergence Structures and Applications to Functional Analysis, chapter 0, pages 119-152, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9942-9_4
    DOI: 10.1007/978-94-015-9942-9_4
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