DF-space

A locally convex topological vector space (TVS) X is a DF-space, also written '(DF)-space', if • 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). == 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, B \cap V is a neighborhood of the origin in B. 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 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. If X is a complete nuclear DF-space then X is a Montel space. == Sufficient conditions ==

The strong dual space X_b^{\prime} of a Fréchet space X 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 (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. • Every Banach space is a DF-space. • 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. The completion of a DF-space is a DF-space. The locally convex sum of a sequence of DF-spaces is a DF-space. An inductive limit of a sequence of DF-spaces is a DF-space. 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. However, An infinite product of non-trivial DF-spaces (i.e. all factors have non-0 dimension) 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. == See also ==