View source for Ptak space

A [[locally convex]] [[topological vector space]] (TVS) <math>X</math> is '''''B''-complete''' or a '''Ptak space''' if every subspace <math>Q \subseteq X^{\prime}</math> is closed in the weak-* topology on <math>X^{\prime}</math> (i.e. <math>X^{\prime}_{\sigma}</math> or <math>\sigma\left(X^{\prime}, X \right)</math>) whenever <math>Q \cap A</math> is closed in <math>A</math> (when <math>A</math> is given the subspace topology from <math>X^{\prime}_{\sigma}</math>) for each equicontinuous subset <math>A \subseteq X^{\prime}</math>.{{sfn|Schaefer|Wolff|1999|p=162}}

''B''-completeness is related to <math>B_r</math>-completeness, where a [[locally convex]] TVS <math>X</math> is '''<math>B_r</math>-complete''' if every {{em|dense}} subspace <math>Q \subseteq X^{\prime}</math> is closed in <math>X^{\prime}_{\sigma}</math> whenever <math>Q \cap A</math> is closed in <math>A</math> (when <math>A</math> is given the subspace topology from <math>X^{\prime}_{\sigma}</math>) for each equicontinuous subset <math>A \subseteq X^{\prime}</math>.{{sfn|Schaefer|Wolff|1999|p=162}}

== Characterizations ==

Throughout this section, <math>X</math> will be a [[Locally convex topological vector space|locally convex]] [[topological vector space]] (TVS). 

The following are equivalent: 
#<math>X</math> is a Ptak space.
#Every continuous [[nearly open]] linear map of <math>X</math> into any locally convex space <math>Y</math> is a topological homomorphism.{{sfn|Schaefer|Wolff|1999|p=163}}
::* A linear map <math>u : X \to Y</math> is called '''nearly open''' if for each neighborhood <math>U</math> of the origin in <math>X</math>, <math>u(U)</math> is dense in some neighborhood of the origin in <math>u(X).</math>

The following are equivalent: 
#<math>X</math> is <math>B_r</math>-complete.
#Every continuous [[biunivocal]], [[nearly open]] linear map of <math>X</math> into any locally convex space <math>Y</math> is a TVS-isomorphism.{{sfn|Schaefer|Wolff|1999|p=163}}

== Properties ==

Every Ptak space is [[Complete topological vector space|complete]].  However, there exist complete Hausdorff [[locally convex]] space that are not Ptak spaces.

{{math theorem|name=Homomorphism Theorem|note=|style=|math_statement=
Every continuous linear map from a Ptak space onto a barreled space is a topological homomorphism.{{sfn|Schaefer|Wolff|1999|p=164}}
}}

Let <math>u</math> be a nearly open linear map whose domain is dense in a <math>B_r</math>-complete space <math>X</math> and whose range is a locally convex space <math>Y</math>. Suppose that the graph of <math>u</math> is closed in <math>X \times Y</math>. If <math>u</math> is injective or if <math>X</math> is a Ptak space then <math>u</math> is an open map.{{sfn|Schaefer|Wolff|1999|p=165}}

== Examples and sufficient conditions ==

There exist B<sub>r</sub>-complete spaces that are not B-complete.  

Every [[Fréchet space]] is a Ptak space.  The strong dual of a [[Reflexive space|reflexive]] Fréchet space is a Ptak space.

Every closed vector subspace of a Ptak space (resp. a B<sub>r</sub>-complete space) is a Ptak space (resp. a <math>B_r</math>-complete space).{{sfn|Schaefer|Wolff|1999|p=162}} and every [[Hausdorff space|Hausdorff]] [[Quotient space (topology)|quotient]] of a Ptak space is a Ptak space.{{sfn|Schaefer|Wolff|1999|p=165}}  
If every Hausdorff quotient of a TVS <math>X</math> is a B<sub>r</sub>-complete space then <math>X</math> is a ''B''-complete space.  

If <math>X</math> is a locally convex space such that there exists a continuous [[nearly open]] surjection <math>u : P \to X</math> from a Ptak space, then <math>X</math> is a Ptak space.{{sfn|Schaefer|Wolff|1999|p=164}}  

If a TVS <math>X</math> has a closed [[hyperplane]] that is B-complete (resp. B<sub>r</sub>-complete) then <math>X</math> is B-complete (resp. B<sub>r</sub>-complete).  

== See also ==

* {{annotated link|Barreled space}}

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

== Bibliography ==

* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn|Husain|Khaleelulla|1978|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->

== External links ==

* [https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]