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Some Special Classes of Locally Convex Spaces

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL,volume 159)

Abstract

If the topology of a locally convex space is the same as the strong topology, the space is said to be barrelled. If the topology is the same as the Mackey topology, and if every linear functional which is bounded on the bounded subsets of the space is continuous, then the space is said to be bornological. Every (F)-space has these properties, and each of these properties entails a sequence of important consequences. The investigation of the properties of barrelled and of bornological spaces, which generalises the theory of (F)-spaces, and which goes back to Mackey and to Bourbaki, forms an important part of the general theory of locally convex spaces. These two classes of spaces are considered in depth in § 27 and § 28.

Keywords

  • Sequence Space
  • Convex Space
  • Double Sequence
  • Strong Topology
  • Closed Linear Subspace

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1983 Springer-Verlag Berlin, Heidelberg

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Köthe, G. (1983). Some Special Classes of Locally Convex Spaces. In: Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften, vol 159. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-64988-2_6

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  • DOI: https://doi.org/10.1007/978-3-642-64988-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64990-5

  • Online ISBN: 978-3-642-64988-2

  • eBook Packages: Springer Book Archive