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I am interested in the Bergman space $A^2$ on the unit disc. According to the Wikipedia article on Bergman spaces, if we have $f(z)= \sum_{n=0}^\infty a_n z^n \in A^2$ then $$\|f\|^2_{A^2} := \frac{1}{\pi} \int_\mathbb{D} |f(z)|^2 \, dz = \sum_{n=0}^\infty \frac{|a_n|^2}{n+1}$$

I can't prove this (last) equality. I'm trying to construct an isometry from this space to the $l^2$ space but I'm running into trouble with double integrals and (infinite) summation.

I also want to prove the part about the reproducing kernel written later in the Wikipedia article, but I believe it will be easier once I set up the isometry above.

Can anyone help me out please? Thanks!

Math Maniac
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The functions $\{z^n\}$ $L^2(\mathbb{D})$ are mutually orthogonal in $L^2(\mathbb{D})$, because $ \int_{\mathbb{D}} z^n \overline z^m =0 $ (integrate in polar coordinates with $r\,d\theta\,dr$, over $\theta$ first).

Also, the integral of $|z^n|^2$ over $\mathbb{D}$ is $\dfrac{2\pi }{2n+2}$, by a similar computation. Hence, $$\|f(z)\|^2_{L^2} = \sum_{n=0}^\infty |a_n|^2 \|z^n\|^2_{L^2} = \frac{1}{\pi}\sum_{n=0}^\infty |a_n|^2 \frac{2\pi }{2n+2}$$ as claimed.