The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}=\langle x,e_{k}\rangle$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $a=(a_{k})_{k\in\mathbb{N}}$ be a weight sequence. Convergence of a series of real numbers and its sum are as defined there. For $k\in\mathbb{N}$, $a_{k}$ is positive, so its multiplicative inverse $a_{k}^{-1}$ exists and is positive (Elementary Order Arithmetic in an Ordered Field §positive-inverse); the sequence $(a_{k}^{-1})_{k\in\mathbb{N}}$ is the sequence of nonnegative weights to which Products and Sums of Weighted Square-Summable Sequences of Real Numbers is applied below.

1. (The noise space)¶ The noise space of $a$ is the set $X^{a}$ of the $x\in X$ for which the series $\sum_{k=1}^{\infty}a_{k}^{-1}x_{k}^{2}$ converges.

2. (The noise pairing and norm)¶ For $x,y\in X^{a}$, the noise pairing $\langle x,y\rangle_{a}$ and the noise norm $|x|_{a}$ are

$$\langle x,y\rangle_{a}=\sum_{k=1}^{\infty}a_{k}^{-1}x_{k}y_{k},\qquad|x|_{a}=\sqrt{\langle x,x\rangle_{a}}\ ;$$

the first series converges by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §products, applied with the weights $a_{k}^{-1}$ in place of its $\mu_{k}$ and the coordinate sequences of $x$ and $y$ in place of its $(a_{k})$ and $(b_{k})$, and $\langle x,x\rangle_{a}=\sum_{k=1}^{\infty}a_{k}^{-1}x_{k}^{2}$ is nonnegative by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, its terms being nonnegative, so that its nonnegative square root $|x|_{a}$ is defined.