Ptak space

A locally convex topological vector space (TVS) formula_1 is "B"-complete or a Ptak space if every subspace formula_2 is closed in the weak-* topology on formula_3 (i.e. formula_4 or formula_5) whenever formula_6 is closed in formula_7 (when formula_7 is given the subspace topology from formula_4) for each equicontinuous subset formula_10.
"B"-completeness is related to formula_11-completeness, where a locally convex TVS formula_1 is formula_11-complete if every subspace formula_2 is closed in formula_4 whenever formula_6 is closed in formula_7 (when formula_7 is given the subspace topology from formula_4) for each equicontinuous subset formula_10.
Characterizations.
Throughout this section, formula_1 will be a locally convex topological vector space (TVS). 
The following are equivalent: 
The following are equivalent: 
Properties.
Every Ptak space is complete. However, there exist complete Hausdorff locally convex space that are not Ptak spaces.
Let formula_22 be a nearly open linear map whose domain is dense in a formula_11-complete space formula_1 and whose range is a locally convex space formula_25. Suppose that the graph of formula_22 is closed in formula_27. If formula_22 is injective or if formula_1 is a Ptak space then formula_22 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 formula_11-complete space). and every Hausdorff quotient of a Ptak space is a Ptak space. 
If every Hausdorff quotient of a TVS formula_1 is a Br-complete space then formula_1 is a "B"-complete space. 
If formula_1 is a locally convex space such that there exists a continuous nearly open surjection formula_35 from a Ptak space, then formula_1 is a Ptak space. 
If a TVS formula_1 has a closed hyperplane that is B-complete (resp. Br-complete) then formula_1 is B-complete (resp. Br-complete). 