DF-space

In the field of functional analysis, a locally convex topological vector space (TVS) X is a DF-space if[1]

X is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of $X^{\prime }$ is equicontinuous), and
X possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets $B_{1},B_{2},\ldots$ such that every bounded subset of X is contained in some $B_{i}$ [2]).

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in (Grothendieck 1954). They also play a considerable part in the theory of topological tensor products.[1] Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If X is a metrizable locally convex space and $V_{1},V_{2},\ldots$ is a sequence of convex 0-neighborhoods in $X_{b}^{\prime }$ such that $V:=\cap _{i}V_{i}$ absorbs every strongly bounded set, then V is a 0-neighborhood in $X_{b}^{\prime }$ (where $X_{b}^{\prime }$ is the continuous dual space of X endowed with the strong dual topology).[3]

Properties

Let X be a DF-space and let V be a convex balanced subset of X. Then V is a neighborhood of the origin if and only if for every convex, balanced, bounded subset $B\subseteq X$ $\text{[math]}$ , $B\cap V$ $\text{[math]}$ is a 0-neighborhood in B.[1]
- Thus, a linear map from a DF-space into a locally convex space is continuous if its restriction to each bounded subset of the domain is continuous.[1]
The strong dual of a DF-space is a Fréchet space.[4]

Sufficient conditions

The strong dual of a metrizable locally convex space is a DF-space (but not conversely, in general).[1] Hence:
- Every normed space is a DF-space.[5]
- Every Banach space is a DF-space.[1]
- Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
Every separated quotient of a DF-space is a DF-space.[4]

Examples

There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.[4] There exist DF-spaces spaces having closed vector subspaces that are not DF-spaces.[6]

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References

Schaefer 1999, pp. 154-155.
Schaefer 1999, p. 25.
Schaefer 1999, pp. 152,154.
Schaefer 1999, pp. 196-197.
Khaleelulla 1982, p. 33.
Khaleelulla 1982, pp. 103-110.

Grothendieck, Alexandre (1954). "Sue les espaces (F) et (DF)". Summa Brasil. Math. 3. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
Grothendieck, Alexandre (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Am. Math. Soc. 16. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
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)
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

DF-space at ncatlab

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[FOOTNOTESchaefer1999154-155-1] Schaefer 1999, pp. 154-155.

[FOOTNOTESchaefer199925-2] Schaefer 1999, p. 25.

[FOOTNOTESchaefer1999152,154-3] Schaefer 1999, pp. 152,154.

[FOOTNOTESchaefer1999196-197-4] Schaefer 1999, pp. 196-197.

[FOOTNOTEKhaleelulla198233-5] Khaleelulla 1982, p. 33.

[FOOTNOTEKhaleelulla1982103-110-6] Khaleelulla 1982, pp. 103-110.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

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 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