On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of degree $n-p$ by $D^{'n-p}(M)$. If we consider the functionals on $D^{'n-p}(M)$ with an approperiate comapctness and continuity assumptions, then what are these functionals? Are they just differential $p$-forms, or could be more then that?


I am asking this question because I want to understand why there is no definition for the pullback of current $T$ in general (as far as I know). Let $f: M\_1 \to M\_2$ be a map between manifolds. Then the pullback $f^\*T$ should be defined as (formally) $$\langle f^\*T, u\rangle = \langle T, f\_\*u \rangle.$$ Here $u$ is a differential $p$-form, in particular, $f\_\*u$ is well-defined as a current under some compactness assumption. Thus the problem is to make sense of $\langle T, f\_\*u \rangle$, i.e. can a current be a functional on the space of current? (Of course, it is enough to have $T$ be a functional on the space $\{f\_\*u\}$, that is why pullback of current is well-defined for submersion maps).



What is the "dual" of the space of currents?
This is a question of functional analysis: Start with a locally convex space $X$, form its dual $X^\*$ consisting of continuous linear functionals on $X$. Now $X^\*$ has several locally convex topologies $\tau$ with the following property:



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> For any $x\in X$ the linear map $L\_x:X^\*\to\mathbb{R}$ is $\tau$-continuous. 
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For such topology $\tau$ we form the bidual $X^{\*\*}\_\tau$ consisting of linear functionals on $X^\*$ that are $\tau$ continuous. From the choice of $\tau$ we deduce that we have a natural map


$$X\ni x\mapsto L\_x\in X^{\*\*}\_\tau. $$


The question is if there exist topologies $\tau$ on $X^\*$ such that the above map is an isomorphism of locally convex spaces.


A natural topology on $X^\*$ is the smallest locally convex topology such that all the linear functionals $L\_x$ are continuous. Let's call this $\tau\_0$. (Traditionally it is denoted by $\sigma(X^\*, X)$.)


We can ask a more refined question: for which locally convex spaces the bidual $X^{\*\*}\_{\tau\_0}$ coincides with $X$. 


There is know a large class of such spaces namely the nuclear spaces. For details see 



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> Gelfand & Shilov: Generalized Functions., vol 2.
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For example the space $X=C^\infty(M)$, $M$ compact has this property. 


