Ptak space explained

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:

X is a Ptak space.

1. 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:

X is B _{r} -complete.

1. 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.

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 x Y . If u is injective or if X is a Ptak space then u is an open map.

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). and every Hausdorff quotient of a Ptak space is a Ptak space. 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.

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).

External links

• Nuclear space at ncatlab