= Solution
A <divisor on an algebraic curve> is a finite integer sum $D=\sum_pn_pp$; its degree is $\sum_pn_p$. For a <meromorphic function> its <principal divisor> records its zero and <pole> orders, and
$$
L(D)=\{0\}\cup\{f\text{ meromorphic}:\operatorname{div}(f)+D\geq0\},\qquad\ell(D)=\dim_{\mathbb C}L(D).
$$
Let $K$ be the divisor of a nonzero meromorphic differential, called a <canonical divisor>. The <Riemann-Roch theorem> states
$$
\boxed{\ell(D)-\ell(K-D)=\deg D+1-g.}
$$
Multiplication by that differential identifies $L(K)$ with the space of holomorphic differentials. By the preceding result $L(0)$ consists exactly of constants, so $\ell(0)=1$. Applying the formula at $D=0$ gives $1-\ell(K)=1-g$, hence \b[$\ell(K)=g$].
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