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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?

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There is a similar question here: mathoverflow.net/questions/42310/when-is-l2x-separable –  Byron Schmuland Nov 15 '10 at 1:17
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up vote 13 down vote accepted

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.

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