The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $h_{k}=\langle h,e_{k}\rangle$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $c$ be a variance sequence.

1. (Cameron-Martin space)¶ The Cameron-Martin space of $c$ is the set $H_{c}$ of those $h\in X$ for which the series $\sum_{k=1}^{\infty}h_{k}^{2}/c_{k}$, whose terms are nonnegative real numbers, converges.

2. (Cameron-Martin square)¶ For $h\in H_{c}$ the Cameron-Martin square of $h$ is the sum of that series, a nonnegative real number as a convergent series of nonnegative terms,

$$|h|_{c}^{2}=\sum_{k=1}^{\infty}\frac{h_{k}^{2}}{c_{k}};$$

the symbol $|h|_{c}^{2}$ is used only as a whole, and no quantity $|h|_{c}$ is defined.