= Solution
Choose a local coordinate $z_i$ at each $p_i$ and map $f\in V$ to all coefficients of the negative-power terms in its Laurent expansions:
$$
f\longmapsto
(a_{i,-1},\ldots,a_{i,-m_i})_{i=1}^n
\in\mathbb C^{\sum_i m_i}.
$$
The kernel consists of globally holomorphic functions on the compact Riemann surface $R$. By the <maximum modulus principle>, every such function is constant, so the kernel has dimension one. Rank--nullity gives
$$
\boxed{\dim_{\mathbb C}V\leq1+\sum_i m_i}.
$$
For $n=0$, this simply says $\dim V=1$.
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