I am studying Peter Major's lecture notes on Multiple Wiener-Ito Integrals.
I have hard time following the arguments of the proofs for the theorem 2.1, which is a first milestone to a Wick polynomial definition and all the other material there...
I feel that this should be pretty straight forward for those who are familiar with this notion...any help would be very appreciated, thanks!
Here is the content of the relevant pages (this is exact exert from pages 6 to 8 of Peter Major's lecture notes):
" [...]
In this section we consider the so-called Wick polynomials, a multi-dimentional generalization of Hermoite-polynomials. they are closely related to multiple Wiener-Ito integral.
Let $X_t, t \in T$ be a set of jointly Gaussian random variables indexed by a parameter set $T$. Let $EX_t = 0$ for all $t \in T$. We define the real Hilbert space $\mathcal{H}_1$ and $\mathcal{H}$ in the following way: A square integrable random variable is in $\mathcal{H}$ if and only if it is measurable with respect to $\sigma$-algebra $\mathcal{B}=\mathcal{B}(X_t, t \in T)$ and the scalar product in $\mathcal{H}$ is defined as $(\xi,\rho)=E\xi\rho, $ for all $\xi,\rho \in \mathcal{H}$.
$\mathcal{H}_1 \subset \mathcal{H}$ is the subspace generated by the finite linear combinations $\sum c_j X_{t_j}$, $t_j \in T$. We consider only such sets $X_t, t \in T$ for which $\mathcal{H}_1$ is separable. $\{X_t, t \in T\}$ can be otherwise arbitrary.
Let $Y_1,Y_2, \dots$ be an orthonormal basis in $\mathcal{H}_1$. then the uncorrelated random variables $Y_1,Y_2, \dots$ are independent, because they are Gaussian. Moreover $\mathcal{B}(Y_1,Y_2, \dots)=\mathcal{B}(X_t, t \in T)$.
Let $H_n$ denote the $n$-th Hermite polynomial with leading coefficient $1$, i.e. let
$ H_n(x) = (-1)^n \exp(\frac{x^2}{2})\frac{d^n}{dx^n}(\exp(\frac{x^2}{2}))) $
We recall the following result from analysis and measure theory:
Theorem $2A$ - The Hermite polynomials $H_n(x), n=0,1,\dots$form a complete orthogonal system in $L_2(\mathbb{R},\mathcal{B},\frac{dx}{\sqrt{2\pi}}\exp(-\frac{x^2}{2}))$ (Here $\mathcal{B}$ denotes a Borel $\sigma$-algebra on the real line).
Let $(X_j,\mathcal{X}_j, \mu_j), j=1,2,\dots$ be countably many copies of a probability space $(X,\mathcal{X}, \mu)$ (We denote the points of $X_j$ by $x_j$). Let $(X^\infty,\mathcal{X}^\infty, \mu^\infty)=\prod_{j=1}^{\infty}(X_j,\mathcal{X}_j, \mu_j)$.
Theorem $2B$ - Let $\phi_0,\phi_1,\dots;\phi_0=1$ be a complete orthonormal system in $L_2(X,\mathcal{X}, \mu)$. Then the functions $\prod_{j=1}^{\infty}\phi_{k_j}(X_j)$ where only finite many indices $k_j$ differ form zero, form a complete orthonormal basis in $(X^\infty,\mathcal{X}^\infty, \mu^\infty)$.
Theorem $2C$ - Let $(X, \mathcal{A})$ be a measurable space $Y_1,Y_2, \dots$ be $\mathcal{A}$-measurable functions such that $\mathcal{B}(Y_1,Y_2, \dots)=\mathcal{A}$. If $\xi$ is an $\mathcal{A}$-measurable functions then there exists an $(\mathbb{R}^\infty, \mathcal{B}^\infty)$ -measurable functions $f$ such that $\xi=f(Y_1,Y_2, \dots)$.
Theorems $2A, 2B, 2C$ have the following important consequence:
Theorem $2.1$ - Let $Y_1,Y_2, \dots$ be othonormal basis in $\mathcal{H}_1$. Then the set of all possible finite products $H_{j_1}(Y_{l_1})\dots H_{j_k}(Y_{l_k})$ is a complete orthogonal system in $\mathcal{H}$.
Proof of Theorem $2.1$
By theorems $2A$ and $2B$, the set of all possible products, $\prod_{j=1}^{\infty}H_{k_j}(x_j)$, where only finite many indeces $k_j$ differ from $0$, is a complete orthogonal system in $L_2(\mathbb{R}^\infty,\mathcal{B}^\infty,\prod_{j=1}^{\infty}\frac{dx_j}{\sqrt{2\pi}}\exp(-\frac{x_{j}^{2}}{2}))$. Since $\mathcal{B}(X_t, t \in T)= \mathcal{B}(Y_1,Y_2, \dots)$, theorem $2C$ implies that the mapping $f(x_1,x_2,\dots)\to f(Y_1,Y_2,\dots)$ is a unitary transform from $L_2(\mathbb{R}^\infty,\mathcal{B}^\infty,\prod_{j=1}^{\infty}\frac{dx_j}{\sqrt{2\pi}}\exp(-\frac{x_{j}^{2}}{2}))$ to $\mathcal{H}$.
Since the image of a complete orthogonal system under a unitary trasformation is again a complete orthonormal system, theorem $2.1$ is proved. [...]"
[1] I don't really understand what is the connection between the first part int the proof that uses theorem $2A,2B$ and the second part, I only know that $ f $ in the second part can be expanded in the $\prod_{j=1}^{\infty}H_{k_j}(x_j)$ since those form an orthogonal systems.
[2] Why is the mapping $f$ is unitary?
[3] What is the intuition behind those claims?
Thank you