In the pages 6 - 7 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
Theorem 1.1.1 states the following:
So, it is clear that $H_n(W_1(x)) = H_n(x)$ is a complete orthogonal system.
However, how do we see that the norm of $H_n(x)$ is 1?
When we look back on the book, we find
So it seems that $\Bbb{E}_\nu[H_n(x) H_n(x)] = \frac{1}{n!}$
In this casem this would be a typo on the book.
Is this the case? or am I missing something?
Edit: The proof seems to work if $Y = X$
Also we might wonder that the result does not follow because $(X,X)$ might not be Gaussian, so it might be good to ask what do we mean by a Gaussian vector: I take the definition from Le Gall (Mouvement Brownian, martingales et calcul stochastique --page 3):
