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 is a
metrizable locally convex space and is a sequence of convex 0-neighborhoods in such that absorbs every strongly bounded set, then is a 0-neighborhood in (where is the continuous dual space of endowed with the strong dual topology).[2]
is a
countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of is equicontinuous), and
possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets such that every bounded subset of is contained in some [3]).
Properties
Let be a DF-space and let be a convex balanced subset of Then is a neighborhood of the origin if and only if for every convex, balanced, bounded subset is a neighborhood of the origin in [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 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:
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 and 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
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]
See also
Barreled space – Type of topological vector spacePages displaying short descriptions of redirect targets
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.
ISBN978-0-8218-1216-7.
MR0075539.
OCLC1315788.
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 is a
metrizable locally convex space and is a sequence of convex 0-neighborhoods in such that absorbs every strongly bounded set, then is a 0-neighborhood in (where is the continuous dual space of endowed with the strong dual topology).[2]
is a
countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of is equicontinuous), and
possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets such that every bounded subset of is contained in some [3]).
Properties
Let be a DF-space and let be a convex balanced subset of Then is a neighborhood of the origin if and only if for every convex, balanced, bounded subset is a neighborhood of the origin in [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 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:
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 and 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
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]
See also
Barreled space – Type of topological vector spacePages displaying short descriptions of redirect targets
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.
ISBN978-0-8218-1216-7.
MR0075539.
OCLC1315788.