A locally convex topological vector space (TVS) is B-complete or a Ptak space if every subspace is closed in the weak-* topology on (i.e. or ) whenever is closed in (when is given the subspace topology from ) 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 (when is given the subspace topology from ) for each equicontinuous subset .
Characterizations
Throughout this section, will be a locally convex topological vector space (TVS).
The following are equivalent: is a Ptak space. Every continuous nearly open linear map of into any locally convex space is a topological homomorphism. A linear map is called nearly open if for each neighborhood of the origin in , is dense in some neighborhood of the origin in
The following are equivalent: is -complete. 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 (resp. a Br-complete space) is a Ptak space (resp. a -complete 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 (resp. Br-complete) then is B-complete (resp. Br-complete).
See also
Notes
References
Bibliography
External links
Nuclear space at ncatlab