DF-space

In the 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. 
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 formula_1 is a metrizable locally convex space and formula_2 is a sequence of convex 0-neighborhoods in formula_3 such that formula_4 absorbs every strongly bounded set, then formula_5 is a 0-neighborhood in formula_3 (where formula_3 is the continuous dual space of formula_1 endowed with the strong dual topology).
Definition.
A locally convex topological vector space (TVS) formula_1 is a DF-space, also written ("DF")-space, if 
Sufficient conditions.
The strong dual space formula_16 of a Fréchet space formula_1 is a DF-space. 
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.