Spectral theorem for compact normal operators

Question

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting $T(x)=A^*A(x)$, $T$ is self-adjoint so its eigenspaces produce H but how can I show it for $A$?