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I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated.

1) It is the intersection of all linear subspaces of full measure.

2) Its elements are those which when seen as linear functionals on the dual of $W$ (say $W^\ast$) are continuous in the inner product induced on $W^\ast$ by $\nu$ (the Gaussian measure).

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The proof of this equivalence can be found in Bogachev: Gaussian Measures, Theorem 2.4.7.

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