Let $W$ be a separable infinite-dimensional Banach space and let $W^*$ denote its continuous dual. Suppose that $\mu$ is a Borel measure on $W$ such that $W^*\subseteq L^2(W,\mu)$. Let $K$ denote the closure of $W^*$ in $L^2(W,\mu)$. What is an example of an element of $K$ that is not in $W^*$? I am especially interested in the special case $W=C([0,1])$ when $\mu$ is Wiener measure.
Some context: Elements of $L^2(W,\mu)$ are square-integrable random variables on $W$, and elements of $W^*$ are random variables represented by bounded linear functionals. In the special case above, elements of $W^*$ are centered Gaussian random variables. $K$ is the dual of the Cameron-Martin space. I believe it consists of all centered Gaussian random variables in $L^2(W,\mu)$, but I am not sure how to construct a Gaussian random variable on $W$ that is not given by a bounded linear functional.