A locally convex topological vector space (TVS) X {\displaystyle X} is B-complete or a Ptak space if every subspace Q ⊆ X ′ {\displaystyle Q\subseteq X^{\prime }} is closed in the weak-* topology on X ′ {\displaystyle X^{\prime }} (i.e. X σ ′ {\displaystyle X_{\sigma }^{\prime }} or σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} ) whenever Q ∩ A {\displaystyle Q\cap A} is closed in A {\displaystyle A} (when A {\displaystyle A} is given the subspace topology from X σ ′ {\displaystyle X_{\sigma }^{\prime }} ) for each equicontinuous subset A ⊆ X ′ {\displaystyle A\subseteq X^{\prime }} . B-completeness is related to B r {\displaystyle B_{r}} -completeness, where a locally convex TVS X {\displaystyle X} is B r {\displaystyle B_{r}} -complete if every dense subspace Q ⊆ X ′ {\displaystyle Q\subseteq X^{\prime }} is closed in X σ ′ {\displaystyle X_{\sigma }^{\prime }} whenever Q ∩ A {\displaystyle Q\cap A} is closed in A {\displaystyle A} (when A {\displaystyle A} is given the subspace topology from X σ ′ {\displaystyle X_{\sigma }^{\prime }} ) for each equicontinuous subset A ⊆ X ′ {\displaystyle A\subseteq X^{\prime }} .
Characterizations Throughout this section, X {\displaystyle X} will be a locally convex topological vector space (TVS). The following are equivalent:
X {\displaystyle X} is a Ptak space. Every continuous nearly open linear map of X {\displaystyle X} into any locally convex space Y {\displaystyle Y} is a topological homomorphism. A linear map u : X → Y {\displaystyle u:X\to Y} is called nearly open if for each neighborhood U {\displaystyle U} of the origin in X {\displaystyle X} , u ( U ) {\displaystyle u(U)} is dense in some neighborhood of the origin in u ( X ) . {\displaystyle u(X).}
The following are equivalent:
X {\displaystyle X} is B r {\displaystyle B_{r}} -complete. Every continuous biunivocal, nearly open linear map of X {\displaystyle X} into any locally convex space Y {\displaystyle 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 {\displaystyle u} be a nearly open linear map whose domain is dense in a B r {\displaystyle B_{r}} -complete space X {\displaystyle X} and whose range is a locally convex space Y {\displaystyle Y} . Suppose that the graph of u {\displaystyle u} is closed in X × Y {\displaystyle X\times Y} . If u {\displaystyle u} is injective or if X {\displaystyle X} is a Ptak space then u {\displaystyle u} is an open map.
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