Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a finite dimensional vector space for any $q$, and in particular the space of global sections $\Gamma(X,E)=H^0(X,E)$ is finite dimensional.
The proof is based on Hodge theory or properties of Frechet spaces.
My question is, if we only consider $X=\mathbb{C}P^n$, could we prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional by a direct, elementary computation?
