DF-space
   HOME

TheInfoList



OR:

In the field of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, DF-spaces, also written (''DF'')-spaces are
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
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 X is a
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
locally convex space and V_1, V_2, \ldots is a sequence of convex 0-neighborhoods in X^_b such that V := \cap_ V_i absorbs every strongly bounded set, then V is a 0-neighborhood in X^_b (where X^_b is the continuous dual space of X endowed with the strong dual topology).


Definition

A
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
(TVS) X is a DF-space, also written (''DF'')-space, if # X is a
countably quasi-barrelled space In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generaliza ...
(i.e. every strongly bounded countable union of equicontinuous subsets of X^ 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


Sufficient conditions

The strong dual space X_b^ of a Fréchet space X is a DF-space.Gabriyelyan, S.S
"On topological spaces and topological groups with certain local countable networks
(2014)
However,


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

* * * * * * *


Citations


Bibliography

* * * * * * *


External links


DF-space at ncatlab
{{TopologicalVectorSpaces Topology Topological vector spaces Functional analysis