Uniform Convergence of Bergman Kernel

Question

In my complex analysis class we've shown that if $\Omega \subset \Bbb C^n$ is compact and if $H^2(\Omega)$ is the set of square integrable holomorphic functions on $\Omega$, then the Bergman kernel is defined as $$K(z,w)=\sum_{j=1}^{\infty}\phi_j(z)\overline{\phi_j(w)}$$ where $\{\phi_j\}_{j=1}^{\infty}$ is an orthonormal basis for $H^2(\Omega)$. However, my professor claimed without proof that the Bergman kernel is uniformly convergent. I'm unable to prove this and every place in the literature I've looked at just states it as a fact, so even a simple proof of this property would be much appreciated!