= Solution
Lift $f$ to a doubly periodic meromorphic function on $\mathbb C$. Integrate it around a fundamental parallelogram whose boundary avoids all poles. Integrals over opposite edges cancel by periodicity, while the <residue theorem> gives
$$
0=\frac1{2\pi i}\int_{\partial P}f(z)\,dz
=\sum_{i=1}^n\operatorname{res}_{p_i}f.
$$
Thus the principal-part map from part (b) takes values in the codimension-one hyperplane on which the sum of residue coefficients is zero. The Mittag--Leffler existence criterion on a compact Riemann surface says that prescribed principal parts occur precisely when their residues pair trivially with every holomorphic one-form. On a complex torus the holomorphic one-forms are the scalar multiples of $dz$, so the single condition is exactly the displayed residue sum. Hence the image has dimension
$\sum_i m_i-1$, and the kernel of constants has dimension one. For $n\geq1$,
$$
\boxed{\dim_{\mathbb C}V=\sum_{i=1}^n m_i}.
$$
Equivalently, this is the genus-one case of the <Riemann-Roch theorem> for a positive divisor.
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