I am reading Lectures on Riemann Surfaces by Otto Forster. He says: (p.110)
The following lemma may be viewed as a generalization of Schwarz's lemma. Let $D,D'$ be a pair of open subsets of $\mathbb{C}$, where $D$ is a relatively compact subset of $D'$. For any $\varepsilon>0$, there is a closed vector space $A\subset L^2(D,\mathcal{O})$, of finite codimension, with $$ \lVert f\rVert_{L^2(D')}\leq \varepsilon \lVert f\rVert_{L^2(D)}$$
He has already shown that $L^2(D,\mathcal{O})$, the space of holomorphic functions on $D$, forms a Hilbert space under the inner product $\iint f\overline{g}dxdy$ thus the "closed" comment.
What does he mean when he says this generalizes Schwarz's lemma? How is this related to Schwarz's lemma?
