The Wiener Chaos Decomposition: Cylindrical Hermite Polynomials Are Dense and the Gaussian Lebesgue Space Is the Orthogonal Sum of the Chaoses

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the cylindrical Hermite span $\mathcal{H}_{\mathrm{fin}}$, the Wiener chaoses $\mathcal{H}_{m}$ and the chaoses up to order $m$ $\mathcal{H}_{\le m}$ ($m\in\mathbb{N}_{0}$) of The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space; sums and series indexed from $0$ are those of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers.

1. (Closed orthogonal subspaces)¶ For every $m\in\mathbb{N}_{0}$, $\mathcal{H}_{m}$ and $\mathcal{H}_{\le m}$ are closed linear subspaces of $L^{2}(\gamma_{c})$, and $\mathcal{H}_{0}$ is the set of classes of constant functions. For $m,m'\in\mathbb{N}_{0}$ with $m\ne m'$, every element of $\mathcal{H}_{m}$ is orthogonal to every element of $\mathcal{H}_{m'}$. For $m\in\mathbb{N}_{0}$ let $J_{m}$ be the orthogonal projection of $L^{2}(\gamma_{c})$ onto $\mathcal{H}_{m}$. Then for every $F\in L^{2}(\gamma_{c})$ and $m\in\mathbb{N}_{0}$, the orthogonal projection of $F$ onto $\mathcal{H}_{\le m}$ is $\sum_{j=0}^{m}J_{j}F$.

2. (Density)¶ $\mathcal{H}_{\mathrm{fin}}$ is dense in $L^{2}(\gamma_{c})$: for every $F\in L^{2}(\gamma_{c})$ and every real $\varepsilon>0$ there is $G\in\mathcal{H}_{\mathrm{fin}}$ with $\lVert F-G\rVert_{2}<\varepsilon$.

3. (Chaos decomposition)¶ For every $F\in L^{2}(\gamma_{c})$, the real sequence $\bigl(\lVert F-\sum_{m=0}^{M}J_{m}F\rVert_{2}\bigr)_{M\in\mathbb{N}}$ converges to $0$, the series $\sum_{m=0}^{\infty}\lVert J_{m}F\rVert_{2}^{2}$ converges, and

$$\lVert F\rVert_{2}^{2}=\sum_{m=0}^{\infty}\lVert J_{m}F\rVert_{2}^{2}.$$