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

Complete weighted Bergman spaces have bounded point evaluations