Space of derivations of holomorphic (analytic) functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:52:18Z http://mathoverflow.net/feeds/question/17227 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17227/space-of-derivations-of-holomorphic-analytic-functions Space of derivations of holomorphic (analytic) functions Lucas Kaufmann 2010-03-05T21:54:55Z 2010-03-05T22:41:55Z <p>Let M be a (real) smooth manifold, $p \in M$ and. The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (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 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).<br> 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.</p> <p>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? <br> 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?)</p>