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 $H^{\star}$, we have the inclusions $$X^{\star} \hookrightarrow H^{\star} \cong H \hookrightarrow X.$$


**My question:** Is $i^{\star} : X^{\star} \hookrightarrow H^{\star}$ a dense injection?



Dense inclusions of Banach spaces and their duals
Yes, if you mean that $i$ is one to one, for an operator $T:X\to Y$ is one to one if and only if $T$\* has weak\* dense range, which means $T$\* has dense range when $X$ is reflexive.


