# Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces

I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, where $X$ is a compact Riemann surface and $\mathcal{O}$ is the sheaf of holomorphic functions. Forster proves this by reducing the problem into considering restriction map of Cech cohomology groups given by a shrinking sequence of relatively compact open coverings(this is done by choosing a proper coordinate patch and using Leray's theorem and Dolbeault's lemma), and introduces $L^2$-norms on such groups. Well, first of all, what's the point in introducing the $L^2$-norm, by which I mean, is that suggesting any interesting topological properties such as compactness or anything else? What's the essential problem lying in this theorem? Secondly, is there any other way of proving this finiteness theorem? I would appreciate your help!

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