In the mathematical 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.
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
absorbs every strongly bounded set, then
endowed with the strong dual topology).
[2] A locally convex topological vector space (TVS)
is a DF-space, also written (DF)-space, if[1] The strong dual space
[7] However, There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.