Ptak space

A locally convex topological vector space (TVS) $$X$$ is B-complete or a Ptak space if every subspace $$Q \subseteq X^{\prime}$$ is closed in the weak-* topology on $$X^{\prime}$$ (i.e. $$X^{\prime}_{\sigma}$$ or $$\sigma\left(X^{\prime}, X \right)$$) whenever $$Q \cap A$$ is closed in $$A$$ (when $$A$$ is given the subspace topology from $$X^{\prime}_{\sigma}$$) for each equicontinuous subset $$A \subseteq X^{\prime}$$.

B-completeness is related to $$B_r$$-completeness, where a locally convex TVS $$X$$ is $$B_r$$-complete if every subspace $$Q \subseteq X^{\prime}$$ is closed in $$X^{\prime}_{\sigma}$$ whenever $$Q \cap A$$ is closed in $$A$$ (when $$A$$ is given the subspace topology from $$X^{\prime}_{\sigma}$$) for each equicontinuous subset $$A \subseteq X^{\prime}$$.

Characterizations
Throughout this section, $$X$$ will be a locally convex topological vector space (TVS).

The following are equivalent:
 * 1) $$X$$ is a Ptak space.
 * 2) Every continuous nearly open linear map of $$X$$ into any locally convex space $$Y$$ is a topological homomorphism.
 * A linear map $$u : X \to Y$$ is called nearly open if for each neighborhood $$U$$ of the origin in $$X$$, $$u(U)$$ is dense in some neighborhood of the origin in $$u(X).$$

The following are equivalent:
 * 1) $$X$$ is $$B_r$$-complete.
 * 2) Every continuous biunivocal, nearly open linear map of $$X$$ into any locally convex space $$Y$$ is a TVS-isomorphism.

Properties
Every Ptak space is complete. However, there exist complete Hausdorff locally convex space that are not Ptak spaces.

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Let $$u$$ be a nearly open linear map whose domain is dense in a $$B_r$$-complete space $$X$$ and whose range is a locally convex space $$Y$$. Suppose that the graph of $$u$$ is closed in $$X \times Y$$. If $$u$$ is injective or if $$X$$ is a Ptak space then $$u$$ 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 $$B_r$$-complete space). and every Hausdorff quotient of a Ptak space is a Ptak space. If every Hausdorff quotient of a TVS $$X$$ is a Br-complete space then $$X$$ is a B-complete space.

If $$X$$ is a locally convex space such that there exists a continuous nearly open surjection $$u : P \to X$$ from a Ptak space, then $$X$$ is a Ptak space.

If a TVS $$X$$ has a closed hyperplane that is B-complete (resp. Br-complete) then $$X$$ is B-complete (resp. Br-complete).