A finite dimensional proof of Verlinde Formula

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

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