Is the space of smooth functions with compact support a DF space?

Question

Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans and Schneiders describe a functor from locally convex spaces to Ind-objects of Banach spaces (sending $E$ to the Ind-object formed by the Gauge-normed spaces generated by absolutely convex bounded subsets ("A topological reconstruction theorem for $D_{\infty}$-modules"). Is there a known good way to describe the image of LFN spaces under this functor? A answer to any of these question would help me immensely - even a good reference.

EDIT: I am interested in showing $Hom_{Ind(ban)}(-,\mathbb{R})$-acyclity for the ind-object of banach spaces associated to the space of test functions - or at least, in computing how close it is to that. This was what the former title was (cryptically) referring to.

Answer 1

The specific space that you mention---smooth functions with compact support---is not a $DF$ space. Under certain situations, an $LF$ space can be a $DFN$ space, e.g., when it is a strict inductive limit of finite dimensional spaces. In general, a (strict) $LF$ space as introduced in the seminal article by Dieudonné and Schwartz is NOT a $DF$ space but it CAN be in special circumstances, e.g., if it is the strict inductive limit of a sequence of Banach spaces. Good references are "Espaces vectorielles topologiques" by Grothendieck (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.

It is not clear to me what the second question of your title means but as regards the one in the text, the space of smooth functions with compact support has, of course, as a complete lcs, a canonical representation as a projective limit of Banach spaces but the seminorms involved are rather complicated. However, the following might be helpful: it is an inductive limit with partition of unity (de Wilde) and this can be used to obtain stronger results than are generally available. For instance, it is not just a quotient of the lcs direct sum of the component $F$ spaces but is even a COMPLEMENTED subspace (Keim, "Induktive und projektive Limiten mit Zerlegung der Einheit", Man. Math. 10(1973) 191-195).