total subset of Bergman space $L_a^2(\mathbb C)$

Question

Let $M$ be a connected open subset in $\mathbb C$ and $a\in M$.As $L_a^2(\mathbb C)$ is Hilbert space and $\varphi(f)=f^n(a)$ is bounded linear functional on $L_a^2(\mathbb C)$ $(n\ge 0)$,according to Riesz representation theorem, we can find $h_n$ such that $\varphi(f)=\langle h_n,f\rangle$.I want to show that $h_n$ is total subset of $L_a^2(\mathbb C)$, but so far I still have no idea about it.I would appreciate if anyone could help.

Answer 1

To show the dependence on $M$ I will denote the Bergman space by $A^2(M)$.

To show that $\{h_n, \, n\geq 0\}$ is a total subset in $A^2(M)$, it is enough to show that its orthogonal is reduced to zero. Take $f \in \{h_n, \, n\geq 0\}^\perp$. This means that for every $n\geq 0$, $f^{(n)}(a) = \langle f,h_n\rangle = 0$. Now since $f$ is analytic it is equal to its Taylor series in a neighbourhood of $a$ which implies that $f$ is zero in a neighbourhood of $a$. Finally since $M$ is connected, $f$ is zero by the identity theorem.