This is from Griffith Harris, p.137:
Let $D$ be a divisor on $M$. If $M$ is compact, for every $D'\in |D|$. There exists $f\in L(D)$ such that $D'= D + (f)$, and conversely any two such functions $f, f'$ differ by a non-zero constant. Thus we have the correspondence $|D|\cong \mathcal{P}(L(D))$.
I wonder where we need the compactness in this proof?
Also "in general, the family of effective divisors on $M$ corresponding to a linear subspace of $\mathcal{P}(H^0(M, \mathcal{O}(L)))$ for some $L\to M$ is called a linear system of divisors. "
I wonder how the correspondences defined in the definition of linear system of divisors, since the case of this correspondence is different from the case of the correspondence defined in $|D|\cong \mathcal{P}(L(D))$.
I think you need the fact that $M$ is compact to show that "$f,f'$ differ by a non-zero constant". Since $(f)=(f')$, $\frac{f}{f'}$ is holomorphic and not identically $0$ on $M$. Because $M$ is compact, $\frac{f}{f'}$ must be a nonzero constant.