bornological topological vector space



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.


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)



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.


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


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