nLab
bornological topological vector space

Contents

Idea

Linear operators on Hilbert spaces are continuous precisely iff they are bounded. A bornological space retains this property 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 about bornological spaces

  • H. H. Schaefer with 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 is in

with review and generalization to bornological abelian groups in

Revised on July 11, 2014 01:47:41 by David Corfield (146.200.7.67)