Here is a claim I found in the proof of lemma 14.3 of Otto Forster's Lectures on Riemann Surfaces. I can't see how this claim is true, and Forster does not explain it. Help please! I have distilled my question down to its basics, i.e. I have removed all other information in the lemma that doesn't pertain to my specific question. Here it is:
Say $D$ is an open subset of $\mathbb{C}$. Let $a_1,...a_n$ be a finite set of points and let $r>0$ be such that $B(a_j,r) \subset D$ for $j=1,\dots, n$. Denote by $L^2(D,\mathcal{O})$ the space of square integrable holomorphic functions on $D$, i.e. holomorphic functions on $D$ for which $$\iint_D |f(x+iy)|^2 \,dx\,dy < \infty$$
The claim is that, if $A \subset L^2(D, \mathcal{O})$ is the subspace of square-integrable holomorphic functions which vanish at all points $a_j$ to at least order $k$, then $A$ has codimension $\leq kn$.
How?? I am finding it difficult to prove.
It's very simple. Define $T:L^2(D,\mathcal O)\to\Bbb C^{n\times k}$ by $$Tf_{j,\alpha}=f^{(\alpha)}(a_j),\quad(1\le j\le n, 0\le\alpha<k).$$Then $A$ is the kernel of $T$.