Ptak space
A locally convex topological vector space (TVS) is B-complete or a Ptak space if every subspace is closed in the weak-* topology on (i.e. or ) whenever is closed in (when is given the subspace topology from ) for each equicontinuous subset .[1]
B-completeness is related to -completeness, where a locally convex TVS is -complete if every dense subspace is closed in whenever is closed in (when is given the subspace topology from ) for each equicontinuous subset .[1]
Characterizations
Throughout this section, will be a locally convex topological vector space (TVS).
The following are equivalent:
- is a Ptak space.
- Every continuous nearly open linear map of into any locally convex space is a topological homomorphism.[2]
- A linear map is called nearly open if for each neighborhood of the origin in , is dense in some neighborhood of the origin in
The following are equivalent:
- is -complete.
- Every continuous biunivocal, nearly open linear map of into any locally convex space 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 be a nearly open linear map whose domain is dense in a -complete space and whose range is a locally convex space . Suppose that the graph of is closed in . If is injective or if is a Ptak space then 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 -complete space).[1] and every Hausdorff quotient of a Ptak space is a Ptak space.[4] If every Hausdorff quotient of a TVS is a Br-complete space then is a B-complete space.
If is a locally convex space such that there exists a continuous nearly open surjection from a Ptak space, then is a Ptak space.[3]
If a TVS has a closed hyperplane that is B-complete (resp. Br-complete) then is B-complete (resp. Br-complete).
See also
- Barreled space – Type of topological vector spacePages displaying short descriptions of redirect targets
Notes
References
- ^ a b c Schaefer & Wolff 1999, p. 162.
- ^ a b Schaefer & Wolff 1999, p. 163.
- ^ a b Schaefer & Wolff 1999, p. 164.
- ^ a b Schaefer & Wolff 1999, p. 165.
Bibliography
- Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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