A finite dimensional proof of Verlinde Formula

We prove two recurrence relations among dimensions $$D_g(r,d,\omega ):=\mathrm{d}\mathrm{i}\mathrm{m}{\mathrm{H}}^0({\mathcal{U}}_{C,\omega },{\mathrm{\Theta }}_{{\mathcal{U}}_{C,\omega }})$$ of spaces of generalized theta functions on moduli spaces ${\mathcal{U}}_{C,\omega }$. By using of these recurrence relations, an explicit formula (Verlinde formula) of $D_g(r,d,\omega )$ is proved (See Theorem 4.3).