The Cameron-Martin Inner Product of Two Cameron-Martin Vectors

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 and $H_{c}$ the Cameron-Martin space of $c$.

(Cameron-Martin inner product)¶ For $h,g\in H_{c}$, the Cameron-Martin inner product of $h$ and $g$ is the sum

$$\langle h,g\rangle_{c}=\sum_{k=1}^{\infty}\frac{h_{k}g_{k}}{c_{k}}$$

of a convergent series. Indeed, put $b_{k}=\frac{1}{2}\bigl(h_{k}^{2}/c_{k}+g_{k}^{2}/c_{k}\bigr)$; the series $\sum_{k}b_{k}$ converges by The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §space and Elementary Properties of Series of Real Numbers §linearity; and $|h_{k}g_{k}|/c_{k}\le b_{k}$, since $c_{k}>0$ by Variance Sequences and Their Truncations §variances and $2|h_{k}g_{k}|\le h_{k}^{2}+g_{k}^{2}$ because $0\le(|h_{k}|-|g_{k}|)^{2}$. So the series converges by An Absolutely Convergent Series of Real Numbers Converges §dominated and An Absolutely Convergent Series of Real Numbers Converges §convergence.