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

Let $H^*$ denote the space of continuous linear functionals on $H$. By the Riesz representation theorem, $$H^* = \{ \langle k, \cdot \rangle : k \in H \}.$$ By the assumption that $\gamma$ has covariance operator $I$, for all $k \in H$, $$\int_H |\langle k, h \rangle|^2 \, \mathrm{d}\gamma(h) = \langle k, k \rangle < \infty,$$ so $H^*$ is contained in the separable Hilbert space $L^2(H)$.

My question: Is $H^*$ dense in $L^2(H)$?

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

Let $H^*$ denote the space of continuous linear functionals on $H$. By the Riesz representation theorem, $$H^* = \{ \langle k, \cdot \rangle : k \in H \}.$$ By the assumption that $\gamma$ has covariance operator $I$, for all $k \in H$, $$\int_H |\langle k, h \rangle|^2 \, \mathrm{d}\gamma(h) = \langle k, k \rangle < \infty,$$ so $H^*$ is contained in the separable Hilbert space $L^2(H)$.

My question: Is $H^*$ dense in $L^2(H)$?

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# If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?

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 $I$.

Let $H^*$ denote the space of continuous linear functionals on $H$. By the Riesz representation theorem, $$H^* = \{ \langle k, \cdot \rangle : k \in H \}.$$ By the assumption that $\gamma$ has covariance operator $I$, for all $k \in H$, $$\int_H |\langle k, h \rangle|^2 \, \mathrm{d}\gamma(h) = \langle k, k \rangle < \infty,$$ so $H^*$ is contained in the separable Hilbert space $L^2(H)$.

My question: Is $H^*$ dense in $L^2(H)$?