If $H$ is a separable Hilbert space, is $L^2(H)$ separable?

Question

Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.

Is the Hilbert space $L^2(H,\gamma)$ separable?

Answer 1

By Example 7.14.13 in Volume 2 of Bogachev's Measure Theory, every Radon measure on $H$ is separable, so that $L^2(H,\gamma)$ is also separable. It is not necessary that $H$ is a Hilbert space, just that every compact subset of $H$ be metrizable.