Suppose I have a separable Banach space $X$. In many texts on infinite dimensional analysis, one considers the set $L^2(\mathbb{P};X)$, $X$ valued random variables that have finite second moment, $\mathbb{E}[\|f\|_X^2]<\infty$. By completion in the sense of equivalence classes of Cauchy sequences, one then has that $L^2(\mathbb{P};X)$ is itself a Banach space.
My question is, if one adds a separability assumption on $X$, can I obtain separability on $L^2(\mathbb{P};X)$? If not, is there a counter-example?
This is not even true when $X=\mathbb R$, without some further assumptions on $(\Omega,\mathcal F,\mathbb P)$. Using the Kolmogorov extension theorem, you can construct an uncountable family of random variables with any given finite dimensional distributions. In particular, you can have an uncountable family of independent coin flips. These random variables have a pairwise positive equal distance in $L^2(\mathbb P;\mathbb R)$, so this space cannot be separable.