I'm trying to prove the following Proposition (see image) in S. Krantz's Book on Bergman Kernal.
I have two questions:
(1) Let $A^2 (\Omega) $ be bergman space. Let $\{\phi_n\}$ be it's complete ON basis.
Fix some $ z \in \Omega $. Does ${\phi_n (z) \in l^2}$ ?
we can show $sup \space {\phi_n (z) } $ is bounded by the following result:
"For $L$ compact subset of $\Omega $ , $ sup \space |f(z)| \leq C_{L} ||f|| $ as z varies over $L$.
Take {z} to be compact set.
But no idea how will we show it's in l2.
(2) How did we reach second step in the proof in the image ? I think the author is using the fact that $ ||a_n|| = sup \{ |<a_n,b_n>| : ||b_n||=1 ,b_n \in l^2\} $ ?
Is that the case ?
But we priori don't know if $a_n \in l^2$ or not?
What is happeing here? Any help is appreciated.
