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

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.

Examples

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

References

Revised on October 20, 2013 09:59:06 by Anonymous Coward (74.68.126.101)