This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, let $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^*$, we have the inclusions $$X^* \hookrightarrow H^* \cong H \hookrightarrow X.$$
My question: Is $i^* : X^* \hookrightarrow H^*$ a dense injection?
This seems like a really simple question, but I'm struggling with it. Let $X$ be a Banach space, let $H$ be a Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^*$, we have the inclusions $$X^* \hookrightarrow H^* \cong H \hookrightarrow X.$$
My question: Is $i^* : X^* \hookrightarrow H^*$ a dense injection?