Let $M$ be a (real) smooth manifold, and $p \in M$.

The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) on the algebra $C^{\infty}(p)$ of differentiable functions defined on some neighbourhood of $p$) is then a $n$-dimensional vector space (this is one way to define the tangent space $T_p M$ after all).

It is easy to see that if we consider instead derivations $D:C(p) \to \mathbb{R}$ on the space $C(p)$ of continuous functions, then the space of derivations is trivial.

My question is: when $M$ is a complex (or analytic) manifold, what is the dimension of the space of derivations on holomorphic (or analytic) functions defined near p?

I've once heard that this space is infinite dimensional. Is this true? (and if it is, is there a simple proof or some reference material?)

`$C^\infty$`

case, the derivations can be seen as a module over`$C^\infty$`

, and this module has finite rank. It may not have a basis, if the manifold doesn't admit a global frame (is that right terminology?), but the finite rank follows by a partition of unity argument. Try to extend that to an analytic manifold, and you run into trouble for lack or partitions of unity. This could be what's behind the problem statement. – Harald Hanche-Olsen Mar 6 '10 at 4:43