Barrelled space

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In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced, absorbing and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

History[edit]

Barrelled spaces were introduced by Bourbaki (1950).

Examples[edit]

Properties[edit]

For a Hausdorff locally convex space X with continuous dual X' the following are equivalent:

  • X is barrelled,
  • every \sigma(X', X)-bounded subset of the continuous dual space X' is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem),[1]
  • for all subsets A of the continuous dual space X', the following properties are equivalent: A is [1]
    • equicontinuous,
    • relatively weakly compact,
    • strongly bounded,
    • weakly bounded,
  • X carries the strong topology \beta(X, X'),
  • every lower semi-continuous semi-norm on X is continuous,
  • the 0-neighborhood bases in X and the fundamental families of bounded sets in E_{\beta}' correspond to each other by polarity.[1]

In addition,

  • Every sequentially complete quasibarrelled space is barrelled.
  • A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.

Quasi-barrelled spaces[edit]

A topological vector space X for which every barrelled bornivorous set in the space is a neighbourhood of 0 is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of X. Every barrelled space is quasi-barrelled.

For a locally convex space X with continuous dual X' the following are equivalent:

  • X is quasi-barrelled,
  • every bounded lower semi-continuous semi-norm on X is continuous,
  • every \beta(X', X)-bounded subset of the continuous dual space X' is equicontinuous.

References[edit]

  1. ^ a b c Schaefer (1999) p. 127, 141, Treves (1995) p. 350