Ptak_space Knowpia

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

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

Characterizations edit

Throughout this section, $X$ will be a locally convex topological vector space (TVS).

The following are equivalent:

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

A linear map $u:X\to Y$ is called nearly open if for each neighborhood $U$ of the origin in $X$ , $u(U)$ is dense in some neighborhood of the origin in $u(X).$

The following are equivalent:

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

Properties edit

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 $u$ be a nearly open linear map whose domain is dense in a $B_{r}$ -complete space $X$ and whose range is a locally convex space $Y$ . Suppose that the graph of $u$ is closed in $X\times Y$ . If $u$ is injective or if $X$ is a Ptak space then $u$ is an open map.^[4]

Examples and sufficient conditions edit

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 $B_{r}$ -complete space).^[1] and 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 $X$ is a locally convex space such that there exists a continuous nearly open surjection $u:P\to X$ from a Ptak space, then $X$ is a Ptak space.^[3]

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).

Notes edit

References edit

^ ^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 edit

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.