Ptak space

A locally convex topological vector space (TVS) ${\displaystyle X}$ is B-complete or a Ptak space if every subspace ${\displaystyle Q\subseteq X^{\prime }}$ is closed in the weak-* topology on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle X_{\sigma }^{\prime }}$ or ${\displaystyle \sigma \left(X^{\prime },X\right)}$) whenever ${\displaystyle Q\cap A}$ is closed in ${\displaystyle A}$ (when ${\displaystyle A}$ is given the subspace topology from ${\displaystyle X_{\sigma }^{\prime }}$) for each equicontinuous subset ${\displaystyle A\subseteq X^{\prime }}$.^[1]

B-completeness is related to ${\displaystyle B_{r}}$-completeness, where a locally convex TVS ${\displaystyle X}$ is ${\displaystyle B_{r}}$-complete if every dense subspace ${\displaystyle Q\subseteq X^{\prime }}$ is closed in ${\displaystyle X_{\sigma }^{\prime }}$ whenever ${\displaystyle Q\cap A}$ is closed in ${\displaystyle A}$ (when ${\displaystyle A}$ is given the subspace topology from ${\displaystyle X_{\sigma }^{\prime }}$) for each equicontinuous subset ${\displaystyle A\subseteq X^{\prime }}$.^[1]

Characterizations

Throughout this section, ${\displaystyle X}$ will be a locally convex topological vector space (TVS).

The following are equivalent:

1. ${\displaystyle X}$ is a Ptak space.
2. Every continuous nearly open linear map of ${\displaystyle X}$ into any locally convex space ${\displaystyle Y}$ is a topological homomorphism.^[2]

• A linear map ${\displaystyle u:X\to Y}$ is called nearly open if for each neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$, ${\displaystyle u(U)}$ is dense in some neighborhood of the origin in ${\displaystyle u(X).}$

The following are equivalent:

1. ${\displaystyle X}$ is ${\displaystyle B_{r}}$-complete.
2. Every continuous biunivocal, nearly open linear map of ${\displaystyle X}$ into any locally convex space ${\displaystyle Y}$ is a TVS-isomorphism.^[2]

Properties

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

Homomorphism Theorem—Every continuous linear map from a Ptak space onto a barreled space is a topological homomorphism.^[3]

Let ${\displaystyle u}$ be a nearly open linear map whose domain is dense in a ${\displaystyle B_{r}}$-complete space ${\displaystyle X}$ and whose range is a locally convex space ${\displaystyle Y}$. Suppose that the graph of ${\displaystyle u}$ is closed in ${\displaystyle X\times Y}$. If ${\displaystyle u}$ is injective or if ${\displaystyle X}$ is a Ptak space then ${\displaystyle u}$ is an open map.^[4]

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 ${\displaystyle B_{r}}$-complete space).^[1] and every Hausdorff quotient of a Ptak space is a Ptak space.^[4] If every Hausdorff quotient of a TVS ${\displaystyle X}$ is a B_r-complete space then ${\displaystyle X}$ is a B-complete space.

If ${\displaystyle X}$ is a locally convex space such that there exists a continuous nearly open surjection ${\displaystyle u:P\to X}$ from a Ptak space, then ${\displaystyle X}$ is a Ptak space.^[3]

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

See also

• Barreled space – Type of topological vector spacePages displaying short descriptions of redirect targets

Notes