THC Science

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.

External links[edit]

Nuclear space at ncatlab

[FOOTNOTESchaeferWolff1999162-1] Schaefer & Wolff 1999, p. 162.

[FOOTNOTESchaeferWolff1999163-2] Schaefer & Wolff 1999, p. 163.

[FOOTNOTESchaeferWolff1999164-3] Schaefer & Wolff 1999, p. 164.

[FOOTNOTESchaeferWolff1999165-4] Schaefer & Wolff 1999, p. 165.

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Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
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