2
$\begingroup$

Let $H$ be a Gaussian Hilbert (containing only centered Gaussian rv) space spanned by the stochastic process $\{\xi_t\}_{t\in T}$. Then for each $\xi\in H$ let $R(\xi)$ be the function on $T$ given by

$$R(\xi)(t)=\langle \xi,\xi_t\rangle_H$$ The Cameron Martin space is the linear space $R(H)=\{R(\xi):\xi\in H\}$.

By application of the Riesz representation theorem, could we conclude that $R(H)$ is a subspace of the dual space $H^*$? Since $\{\xi_t\}_{t\in T}$ is total on $H$ could we conclude that $R(H)$ will be a dense subspace?

$\endgroup$

1 Answer 1

2
$\begingroup$

$R(\xi)$ is a function of $t$ and not a linear functional on $H$ so $R(H)$ is not a subspace of $H^*$. The only linear functionals here are the ones given by $R(\cdot)(t): H \to \mathbb{R}$ for fixed values of $t$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .