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Let M be a (real) smooth manifold, $p \in M$ and$U$ a neighborhood of $p$. . The space of (linear) derivations $D:C^{\infty}(U) D:C^{\infty}(p) \to \mathbb{R}$ (ie, 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 U 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 $D:C(U) D:C(p) \to \mathbb{R}$ on continuos 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 on Udefined near p?
I've once heard that this space is infinite dimensional. Is this true? (and if it is there's a simple proof or some reference material?)

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# Space of derivations of holomorphic (analytic) functions

Let M be a (real) smooth manifold, $p \in M$ and $U$ a neighborhood of $p$. The space of (linear) derivations $D:C^{\infty}(U) \to \mathbb{R}$ on differentiable functions defined on U 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 $D:C(U) \to \mathbb{R}$ on continuos 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 on U?
I've once heard that this space is infinite dimensional. Is this true? (and if it is there's a simple proof or some reference material?)