Ptak space

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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].[1] 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].[1]

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.[2]
  • 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.[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 [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.[4]

Examples and sufficient conditions

There exist Br-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 Br-complete space) is a Ptak space (resp. a [math]\displaystyle{ B_r }[/math]-complete space).[1] and every Hausdorff quotient of a Ptak space is a Ptak space.[4] If every Hausdorff quotient of a TVS [math]\displaystyle{ X }[/math] is a Br-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.[3]

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

See also

Notes

References

Bibliography

External links