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 X ′ {\displaystyle X^{\prime }} {\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:
X is a Ptak space.
Every continuous nearly open linear map of X into any locally convex space Y is a topological homomorphism.[2]
Recall that a linear map \(u:X\to Y \) is called nearly open if for each neighborhood U of the origin in \( {\displaystyle u(U)} \) is dense in some neighborhood of the origin in \( {\displaystyle u(X)}. \)
The following are equivalent:
X is \( {\displaystyle B_{r} \)}-complete.
Every continuous biunivocal, nearly open linear map of X into any locally convex space Y 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 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 \) 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 Fréchet space is a Ptak space.
The strong dual of a relvexive Fréchet space is a Ptak space.
Every closed subspace of a Ptak space (resp. a Br-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 Br-complete space then X is a B-complete space.
If a TVS X has a closed hyperplane that is B-complete (resp. Br-complete) then X is B-complete (resp. Br-complete).
There are Br-complete spaces that are not B-complete.
See also
Barreled spaces
References
Schaefer & Wolff 1999, p. 162.
Schaefer & Wolff 1999, p. 163.
Schaefer & Wolff 1999, p. 164.
Schaefer & Wolff 1999, p. 165.
Bibliography
Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
Khaleelulla, S. M. (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.
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.
Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
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.
Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
External links
Nuclear space at ncatlab
Topological vector spaces (TVSs)
Basic concepts
Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results
Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps
Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear 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 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 (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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