Ptak space

A locally convex topological vector space (TVS) X is B-complete or a Ptak space if every subspace ${\displaystyle Q\subseteq X^{\prime }}$ is closed in the weak-* topology on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle X_{\sigma }^{\prime }}$ or ${\displaystyle \sigma \left(X^{\prime },X\right)}$) whenever ${\displaystyle Q\cap A}$ is closed in A (when A is given the subspace topology from ${\displaystyle X_{\sigma }^{\prime }}$) for each equicontinuous subset ${\displaystyle A\subseteq X^{\prime }}$.[1]

B-completeness is related to ${\displaystyle B_{r}}$-completeness, where a locally convex TVS X is ${\displaystyle B_{r}}$-complete if every dense subspace ${\displaystyle Q\subseteq X^{\prime }}$ is closed in ${\displaystyle X_{\sigma }^{\prime }}$ whenever ${\displaystyle Q\cap A}$ is closed in A (when A is given the subspace topology from ${\displaystyle X_{\sigma }^{\prime }}$) for each equicontinuous subset ${\displaystyle A\subseteq X^{\prime }}$.[1]

Characterizations

Let X be a locally convex TVS. Then the following are equivalent:

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

• Recall that a linear map ${\displaystyle u:X\to Y}$ is called nearly open if for each neighborhood U of the origin in X, ${\displaystyle u(U)}$ is dense in some neighborhood of the origin in ${\displaystyle u(X)}$.

The following are equivalent:

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

Properties

• (Homomorphism Theorem) Every continuous linear map from a Ptak space onto a barreled space is a topological homomorphism.[3]
• Let ${\displaystyle u}$ be a nearly open linear map whose domain is dense in a ${\displaystyle B_{r}}$-complete space X and whose range is a locally convex space Y. Suppose that the graph of u is closed in ${\displaystyle X\times Y}$. If u is injective or if X is a Ptak space then u is an open map.[4]

Examples and sufficient conditions

• Every closed subspace of a Ptak space (resp. a B_r-complete space) is a Ptak space (resp. a ${\displaystyle B_{r}}$-complete space).[1]
• If X is a locally convex space such that there exists a continuous nearly open surjection u : P → X from a Ptak space, then X is a Ptak space.[3]
• Every Hausdorff quotient of a Ptak space is a Ptak space.[4]
• If every Hausdorff quotient of a TVS X is a B_r-complete space then X is a B-complete space.
• If a TVS X has a closed hyperplane that is B-complete (resp. B_r-complete) then X is B-complete (resp. B_r-complete).

There are B_r-complete spaces that are not B-complete.

See also

• Barreled spaces

References

1. Schaefer 1999, p. 162. sfn error: no target: CITEREFSchaefer1999 (help)
2. Schaefer 1999, p. 163. sfn error: no target: CITEREFSchaefer1999 (help)
3. Schaefer 1999, p. 164. sfn error: no target: CITEREFSchaefer1999 (help)
4. Schaefer 1999, p. 165. sfn error: no target: CITEREFSchaefer1999 (help)

• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061.CS1 maint: ref=harv (link)
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.CS1 maint: ref=harv (link)
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.CS1 maint: ref=harv (link)

External links

• Nuclear space at ncatlab