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In the mathematical field of functional analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products.^[1]

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in (Grothendieck 1954). 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).^[2]

Definition[edit]

A locally convex topological vector space (TVS) $X$ is a DF-space, also written (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 $B_{i}$ ^[3]).

Properties[edit]

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 $B\subseteq X,$ $B\cap V$ is a neighborhood of the origin in $B.$ ^[1] Consequently, 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 space of a DF-space is a Fréchet space.^[4]
Every infinite-dimensional Montel DF-space is a sequential space but not a Fréchet–Urysohn space.
Suppose $X$ is either a DF-space or an LM-space. If $X$ is a sequential space then it is either metrizable or else a Montel space DF-space.
Every quasi-complete DF-space is complete.^[5]
If $X$ is a complete nuclear DF-space then $X$ is a Montel space.^[6]

Sufficient conditions[edit]

The strong dual space $X_{b}^{\prime }$ of a Fréchet space $X$ is a DF-space.^[7]

The strong dual of a metrizable locally convex space is a DF-space^[8] but the convers is in general not true^[8] (the converse being the statement that every DF-space is the strong dual of some metrizable locally convex space). From this it follows:
- Every normed space is a DF-space.^[9]
- Every Banach space is a DF-space.^[1]
- Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
Every Hausdorff quotient of a DF-space is a DF-space.^[10]
The completion of a DF-space is a DF-space.^[10]
The locally convex sum of a sequence of DF-spaces is a DF-space.^[10]
An inductive limit of a sequence of DF-spaces is a DF-space.^[10]
Suppose that $X$ and $Y$ are DF-spaces. Then the projective tensor product, as well as its completion, of these spaces is a DF-space.^[6]

However,

An infinite product of non-trivial DF-spaces (i.e. all factors have non-0 dimension) is not a DF-space.^[10]
A closed vector subspace of a DF-space is not necessarily a DF-space.^[10]
There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS.^[10]

Examples[edit]

There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.^[10] There exist DF-spaces having closed vector subspaces that are not DF-spaces.^[11]

Citations[edit]

^ ^a ^b ^c ^d ^e Schaefer & Wolff 1999, pp. 154–155.
^ Schaefer & Wolff 1999, pp. 152, 154.
^ Schaefer & Wolff 1999, p. 25.
^ Schaefer & Wolff 1999, p. 196.
^ Schaefer & Wolff 1999, pp. 190–202.
^ ^a ^b Schaefer & Wolff 1999, pp. 199–202.
^ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
^ ^a ^b Schaefer & Wolff 1999, p. 154.
^ Khaleelulla 1982, p. 33.
^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 196–197.
^ Khaleelulla 1982, pp. 103–110.

Bibliography[edit]

Grothendieck, Alexander (1954). "Sur les espaces (F) et (DF)". Summa Brasil. Math. (in French). 3: 57–123. MR 0075542.
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
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.
Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
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.
Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.

External links[edit]

DF-space at ncatlab

[FOOTNOTESchaeferWolff1999154–155-1] Schaefer & Wolff 1999, pp. 154–155.

[FOOTNOTESchaeferWolff1999152,_154-2] Schaefer & Wolff 1999, pp. 152, 154.

[FOOTNOTESchaeferWolff199925-3] Schaefer & Wolff 1999, p. 25.

[FOOTNOTESchaeferWolff1999196-4] Schaefer & Wolff 1999, p. 196.

[FOOTNOTESchaeferWolff1999190–202-5] Schaefer & Wolff 1999, pp. 190–202.

[FOOTNOTESchaeferWolff1999199–202-6] Schaefer & Wolff 1999, pp. 199–202.

[Gabriyelyan_2014-7] Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

[FOOTNOTESchaeferWolff1999154-8] Schaefer & Wolff 1999, p. 154.

[FOOTNOTEKhaleelulla198233-9] Khaleelulla 1982, p. 33.

[FOOTNOTESchaeferWolff1999196–197-10] ^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 196–197.

[FOOTNOTEKhaleelulla1982103–110-11] Khaleelulla 1982, pp. 103–110.

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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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