= Solution
Choose a nonzero <meromorphic differential on a Riemann surface> $\eta$ and write $K=(\eta)$. Multiplication defines a linear map
$$
\mathcal L(K)\longrightarrow\operatorname{HD}(C),\qquad f\longmapsto f\eta.
$$
It takes values in the <holomorphic differential forms> because $(f\eta)=(f)+K\geq0$. It is injective since $\eta\ne0$. It is surjective because, for any <holomorphic differential form> $\omega$, the ratio $\omega/\eta$ is a <meromorphic function> and has $(\omega/\eta)+K=(\omega)\geq0$. Thus this map identifies $\operatorname{HD}(C)$ with the <canonical Riemann-Roch space>.
Furthermore, $\mathcal L(0)=\mathbb C$: a <meromorphic function> with no <poles> is <holomorphic> on the <compact Riemann surface> $C$, and hence constant by the <maximum modulus principle>. Applying the <Riemann-Roch theorem> with $D=0$ gives
$$
1-\ell(K)=1-g.
$$
Therefore
$$
\boxed{\dim_{\mathbb C}\operatorname{HD}(C)=\ell(K)=g.}
$$
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