Definition 1.1.7 (Classical Cameron-Martin Space).label Let $\ccm$ be the set of absolutely continuous functions $\theta: [0, \infty) \to \real^{d}$ such that $\theta(0) = 0$ and $D\theta \in L^{2}([0, \infty); \real^{d})$, equipped with the inner product
\[\angles{\theta, \eta}_{\ccm}= \angles{D\theta, D\eta}_{L^2([0, \infty); \real^d)}\quad \forall \theta, \eta \in \ccm\]
then
- (1)
For each $\theta \in \ccm$, $\norm{\theta}_{\sps}\le \frac{1}{2}\norm{\theta}_{\ccm}$.
- (2)
$\ccm$ is a dense subspace of $\sps$.
Thus $\ccm$ is continuously embedded in $\sps$ as a dense subspace, known as the classical Cameron-Martin space for the classical Wiener measure.
Proof. $(1)$: Let $\theta \in \ccm$, then for each $t \ge 0$,
\[\abs{\theta(t)}= \abs{\int_{[0, \infty)} \one_{[0, t)} D\theta ds}\le \normn{\one_{[0, t)}}_{L^2([0, \infty))}\normn{D\theta}_{L^2([0, \infty); \real^d)}= \sqrt{t}\norm{\theta}_{\ccm}\]
Thus
\[\frac{\abs{\theta(t)}}{1 + t}\le \norm{\theta}_{\ccm} \frac{\sqrt{t}}{1 + t}\le \frac{\norm{\theta}_{\ccm}}{2}\]
$(2)$: Follows from the density of $C_{c}^{\infty}(\real^{d})$ in $C_{0}((0, \infty); \real^{d})$ and thus in $\sps$.$\square$