I am reading about the Dirichlet Space right now. The definition of a Dirichlet space is the set of all holomorphic functions in the unit disc that are finite with respect to the semi-norm: $\mid \mid f \mid \mid$ = $\int _D \mid f^' (z)\mid ^2 dA(z)$
Here $dA$ is the normalized area measure, because we define $dA(x+iy)= \frac 1 \pi dx dy$. In the complex case, homorphicity and analyticity are equivalent, so we can say $f(z)=\sum^\infty_0a_nz^n$, then because $dA$ is an area normalizing measure, we can say $\mid \mid f \mid \mid$ = $\int _D \mid f^' (z)\mid ^2 dA(z)$ = $\int _D \mid f^' (z)\mid ^2 dA(z)$ = $\int_D \mid \sum_1^\infty na_n z^n-1 \mid^2 dA(z)$ = $\sum_1^\infty n\mid a_n \mid^2$. Question: Why is the last equality true?