Disclaimer: I hope but do not guarantee I am using correct terminology!
Let $M$ be a compact complex manifold, and let $D$ be an effective divisor (by this I mean a finite formal sum over positive integers of irreducible closed hypersurfaces of $M$).
Let $E$ be an analytic vector bundle over $M$, and let $S$ be the (complex) vector space of meromorphic sections $s$ of $E$ with poles at worst $D$ (ie $\mathrm{Div}(s) + D \geq 0$). Is it true that $S$ has finite dimension?