I have two related questions:
Is $L^2(X,\mu)$ separable if $\mu$ is a probability measure that is Lebesgue except it may have finitely many atoms?
If I want to prove that a function is continuous in the weak topology of $L^2$ is it enough to show that it is sequentially continuous?
If the answer to the first question is yes, the second one would follow because then the $L^2$ space would be metrizable.
Thanks!