Let $H$ be a separable Hilbert space and $\nu$ its canonical cylinder measure. By the construction of Gross there exists a separable Banach space $X$ s.t. $i: H \hookrightarrow X$ and a measure on $X$ defined by
$$\mu(C) = \nu(i^{-1}(C)) = \nu(C \cap H), \forall C \in \mathcal{C} $$
where $\mathcal{C}$ denotes the cylinder algebra on $X$, and the $i(H)$ coincides with the Cameron-Martin subspace of $(X, \mu)$.
Now, on the one hand, the definition of the measure $\mu$ implies that
$$\mu(H) = \nu(H \cap H) = \nu(H) = 1$$
be the definition of being a cylinder measure. But on the other hand, one can show that the Cameron-Martin subspace has measure $0$ unless $X$ is finite-dimensional.
Where is my mistake here?
When $X$ is infinite-dimensional, the Cameron-Martin space $H$ is not a cylinder set of $X$ (exercise), and so the first equation does not imply $\mu(H)=1$, and there is no contradiction.