The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, the coordinate maps $\pi_{1},\pi_{2}:X\times X\to X$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs and the Borel $\sigma$-algebras of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, let $a=(a_{k})_{k\in\mathbb{N}}$ be a weight sequence and let $\bar{a}\in\mathbb{R}$ satisfy $a_{k}\le\bar{a}$ for every $k\in\mathbb{N}$, a real bound, which exists by that definition. For $k\in\mathbb{N}$, $a_{k}^{1/2}$ is the positive square root of $a_{k}$ and $a_{k}^{-1/2}$ is its multiplicative inverse. Let $X^{a}$ be the noise space of $a$, with the inner product $\langle\cdot,\cdot\rangle_{a}$ and norm $|\cdot|_{a}$ of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product. For $N\in\mathbb{N}$ let $S_{N}:X\to\mathbb{R}$ be the function $S_{N}(x)=\sum_{k=1}^{N}a_{k}^{-1}x_{k}^{2}$. Square roots are those of Existence and Uniqueness of the Nonnegative Square Root. Then the following hold.

1. (A real Hilbert space)¶ $X^{a}$ is a linear subspace of $X$: it contains $0_{X}$, and $x+y\in X^{a}$ and $tx\in X^{a}$ for all $x,y\in X^{a}$ and $t\in\mathbb{R}$. With the vector operations of $X$ restricted to $X^{a}$ and the inner product $\langle\cdot,\cdot\rangle_{a}$, $X^{a}$ is a real Hilbert space with zero vector $0_{X}$, and $|\cdot|_{a}$ is the norm of its inner product.

2. (Orthonormal basis)¶ For $k\in\mathbb{N}$ let $f_{k}=a_{k}^{1/2}e_{k}$. Then $f_{k}\in X^{a}$, the sequence $(f_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X^{a}$, and $\langle x,f_{k}\rangle_{a}=a_{k}^{-1/2}x_{k}$ for every $x\in X^{a}$ and $k\in\mathbb{N}$.

3. (Continuous embedding)¶ For every $x\in X^{a}$ one has $|x|^{2}\le\bar{a}\,|x|_{a}^{2}$.

4. (Partial sums)¶ Every $S_{N}$ is continuous from $(X,d)$ to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line, and $S_{N}\le S_{N+1}$ on $X$. A vector $x\in X$ belongs to $X^{a}$ if and only if the sequence $(S_{N}(x))_{N\in\mathbb{N}}$ is bounded above, and then $|x|_{a}^{2}=\sup_{N\in\mathbb{N}}S_{N}(x)$.

5. (Closed balls)¶ For every $R\in\mathbb{R}$ with $0\le R$, the set $\{x\in X^{a}:|x|_{a}\le R\}$ is closed in $(X,d)$.

6. (Borel measurability)¶ $X^{a}$ belongs to $\mathcal{B}(X)$, and the function $n_{a}:X\to\mathbb{R}$ with $n_{a}(x)=|x|_{a}^{2}$ for $x\in X^{a}$ and $n_{a}(x)=0$ for $x\in X\setminus X^{a}$ is Borel.

7. (Differences of pairs)¶ The set $D_{a}=\{z\in X\times X:\pi_{2}(z)-\pi_{1}(z)\in X^{a}\}$ belongs to $\mathcal{B}(X\times X)$, and the function $c_{a}:X\times X\to\mathbb{R}$, $c_{a}(z)=n_{a}(\pi_{2}(z)-\pi_{1}(z))$, is Borel and nonnegative.

8. (Measurable maps into the noise space)¶ Let $(S,\mathcal{S})$ be a measurable space, measurability being that of Measurable Function and Real-Valued Measurable Function, with the Borel $\sigma$-algebra of $X^{a}$ for the distance of $|\cdot|_{a}$. A map $v:S\to X^{a}$ is measurable if and only if, for every $k\in\mathbb{N}$, the function $s\mapsto\langle v(s),e_{k}\rangle$ is measurable. In particular, if $v:S\to X$ is measurable and $v(s)\in X^{a}$ for every $s\in S$, then $v$ is measurable as a map into $X^{a}$.