DF-space
In the mathematical field of functional analysis, DF-spaces, also written -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.
DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in.
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 .
Definition
A locally convex topological vector space is a DF-space, also written -space, if- is a countably quasi-barrelled space, and
- possesses a fundamental sequence of bounded.
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 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.
- The strong dual space of a DF-space is a Fréchet space.
- Every infinite-dimensional Montel DF-space is a sequential space but a Fréchet–Urysohn space.
- Suppose is either a DF-space or an LM-space. If is a sequential space then it is either metrizable or else a Montel space DF-space.
- Every quasi-complete DF-space is complete.
- If is a complete nuclear DF-space then is a Montel space.
Sufficient conditions
The strong dual space of a Fréchet space is a DF-space.- The strong dual of a metrizable locally convex space is a DF-space but the convers is in general not true. From this it follows:
- Every normed space is a DF-space.
- Every Banach space is a DF-space.
- Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
However,
- An infinite product of non-trivial DF-spaces is a DF-space.
- A closed vector subspace of a DF-space is not necessarily a DF-space.
- There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS.
Examples
There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.There exist DF-spaces having closed vector subspaces that are not DF-spaces.