  Let $\Omega\subset \mathbb{C}$ be an arbitrary domain in the one-dimensional
complex plane equipped with a positive Radon measure $\mu$. For any $1\le p<
\infty$, it is shown that the weighted Bergman space $A^p(\Omega, \mu)$ of
holomorphic functions is a Banach space if and only if $A^p(\Omega, \mu)$ has
locally uniformly bounded point evaluations. In particular, in the case $p =2$,
any complete Bergman space $A^2(\Omega, \mu)$ is automatically a reproducing
kernel Hilbert space.
