$\newcommand{\Der}{\operatorname{Der}}$ $\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential geometry in my mathematical toolkit. In any case, it is the toy case of something I need in a stalled bit of research. If you think the question belongs elsewhere then let me know.)
Let $M$ be a compact connected smooth oriented manifold -- I have in mind the case of a compact connected Lie group, but would like to avoid using the group structure for now. Let $A=C^\infty(M)$ with its usual Fréchet algebra structure, and let $A'$ denote its strong dual. I am trying to understand the space $\Der(A,A')$ of continuous derivations $A\to A'$; more concretely, the space of all continuous bilinear forms $D: A\times A \to\Real$ which satisfy $$ D(fg)(h)=D(g)(hf)+D(f)(gh) \quad\hbox{for all $f,g,h\in A$.} $$
In particular: given $D$ as above, I would like to show there exists a continuous derivation $\widetilde{D}: A\to A$ and some $\Psi\in A'$ such that $$ D(a)(b) = \Psi(b\widetilde{D}(a)) \quad\hbox{for all $a,b\in A$.} $$
I suspect this can be deduced from e.g. the calculation of Hochschild cohomology groups of $A$ given in Lemma 45 (p.128) of Connes's NCDG paper -- but it seems that for degree 1 cohomology one ought to have a more hands-on approach, and ultimately it is the proof rather than the result which I want to get a grip on. In particular, is there an argument which doesn't need the full machinery of the de Rham complex, projective resolutions in suitable categories of topological modules, etc?
The natural approach is to try and reduce the problem down to local co-ordinates, then use e.g. approximation by Taylor series; but I'm finding it tough to get the reduction to work rigorously, and suspect I am reinventing the wheel with unnecessary and unhelpful corners on it. Hence, I would be grateful if someone could either
- give a self-contained reference, preferably using relatively low-level tools
- give me some pointers as to how a proof would go
- give a counter-example if the desired result is actually false.
P.S. I seem to remember there was an earlier MO question asking about the Kähler module for $A$, which while not exactly the same question (owing to continuity requirements here) probably had some relevant comments and answers.
