Ptak space

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

Characterizations

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

The following are equivalent:

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

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

The following are equivalent:

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

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.

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

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

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

Bibliography

• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 

External links

• Nuclear space at ncatlab