nLab bornological topological vector space

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Contents

Contents

Idea

Bornological vector spaces are a kind of topological vector spaces.

On normed spaces linear operators are continuous iff they are bounded. On bornological spaces this property is retained by definition.

The discussion below is about bornological CVSes, but there is a more general notion of bornological space.

Definition

A locally convex topological vector space EE is bornological if every circled, convex subset AEA \subset E that absorbs every bounded set in EE is a neighbourhood of 00 in EE. Equivalently, every seminorm that is bounded on bounded sets is continuous.

The bornology of a given TVS is the family of bounded subsets.

Given a locally convex TVS EE with initial topology T 0T_0, there is a finest topology TT such that the family of bounded subsets of TT coincides with T 0T_0. The space EE equipped with the topology TT is called the bornologification of EE, or the bornological space associated with (E,T 0)(E, T_0)

Properties

Maps on bornological spaces

Theorem

Let UU be a linear map from a bornological space EE to any locally convex TVS, then the following statements are equivalent:

  • UU is continuous,

  • UU is bounded on bounded sets,

  • UU maps null sequences to null sequences.

Relation to Banach spaces

Every inductive limit of Banach spaces is a bornological vector space. (Alpay-Salomon 13, prop. 2.3)

Conversely, every bornological vector space is an inductive limit of normed spaces, and of Banach spaces if it is quasi-complete (Schaefer-Wolff 99)

Examples

Every metrizible locally convex space is bornological, that is every Fréchet space and thus every Banach space.

References

  • Wikipedia, Bornological spaces

  • H. H. Schaefer, M. P. Wolff, section 8 of: Topological vector spaces, Springer (1999)

  • Daniel Alpay, Guy Salomon, On algebras which are inductive limits of Banach spaces (arXiv:1302.3372)

Discussion of bornological vector spaces forming a quasi-abelian category:

with review and generalization to bornological abelian groups:

See also:

Last revised on May 1, 2024 at 03:39:10. See the history of this page for a list of all contributions to it.