This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ 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?