Ptak space


A locally convex topological vector space is B-complete or a Ptak space if every subspace is closed in the weak-* topology on whenever is closed in for each equicontinuous subset.
B-completeness is related to -completeness, where a locally convex TVS is -complete if every subspace is closed in whenever is closed in for each equicontinuous subset.

Characterizations

Throughout this section, will be a locally convex topological vector space.
The following are equivalent:
  1. is a Ptak space.
  2. Every continuous nearly open linear map of into any locally convex space is a topological homomorphism.
The following are equivalent:
  1. is -complete.
  2. Every continuous biunivocal, nearly open linear map of into any locally convex space is a TVS-isomorphism.

Properties

Every Ptak space is complete. However, there exist complete Hausdorff locally convex space that are not Ptak spaces.
Let be a nearly open linear map whose domain is dense in a -complete space and whose range is a locally convex space. Suppose that the graph of is closed in. If is injective or if is a Ptak space then is an open map.

Examples and sufficient conditions

There exist Br-complete spaces that are not B-complete.
Every Fréchet space is a Ptak space. The strong dual of a reflexive Fréchet space is a Ptak space.
Every closed vector subspace of a Ptak space is a Ptak space. and every Hausdorff quotient of a Ptak space is a Ptak space.
If every Hausdorff quotient of a TVS is a Br-complete space then is a B-complete space.
If is a locally convex space such that there exists a continuous nearly open surjection from a Ptak space, then is a Ptak space.
If a TVS has a closed hyperplane that is B-complete then is B-complete.