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. $X_{\sigma }^{\prime }$ or $\sigma (X^{\prime },X)$) whenever $Q\cap A$ is closed in $A$ (when $A$ is given the subspace topology from $X_{\sigma }^{\prime }$) for each equicontinuous subset $A\subseteq X^{\prime }$.Template:Sfn

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

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.Template:Sfn

• 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.Template:Sfn

Properties

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

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.Template:Sfn

Examples and sufficient conditions

There exist B_r-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 B_r-complete space) is a Ptak space (resp. a $B_r$-complete space).Template:Sfn and every Hausdorff quotient of a Ptak space is a Ptak space.Template:Sfn If every Hausdorff quotient of a TVS $X$ is a B_r-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.Template:Sfn

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

See also

• Template:Annotated link

Bibliography

• Template:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces

External links

• Nuclear space at ncatlab