I am starting to read about the Hida White Noise theory, and honestly there are some definitions/concepts that are not totally clear to me.
Start by assuming that $S(\mathbb R)$ is the space of rapidly decreasing functions and $S'(\mathbb R)$ denotes its dual.
It's easy to see that $S\subset L^2\subset S'$.
Then we define the "White noise probability space" as the triple $(S',\mathcal B(S'),\mu)$ where $\mathcal B$ stands for the Borel sigma algebra and $\mu$ is a generalized Gaussian measure (obtained by applying the Bochner-Minlos theorem).
At this point I would like to obtain two spaces such that the following holds
$$V\subset L^2(S')\subset V'.$$
This is basically what Hida did with his test function and distribution spaces, but I've seen many ways to construct them and I have some doubts regarding a particular one.
In this set of notes the author introduces this spaces by firstly defining
$(S)_p=\{\phi\in (L^2):\|\phi\|_p<\infty\}$ where $$\|\phi\|_p=\left(\sum_{n=0}^{\infty} \|(A^p)^{\otimes n} f_n\|_{L^2(\mathbb R^n)}^2\right)^{1/2}, p\geq 0.$$ Where we used the chaotic representation $\phi=\sum_n I_n(f_n), f_n\in \hat{L}^2(\mathbb R^n)$ and the differential operator is defined as $A= (-\Delta +x^2+1)$.
Now my question is, what's the justification for using this norm? From where does this differential operator come from?
I understand that (when defining the space of test functions) we need to build a "stronger" norm, but I don't see how we arrive to this particular construction of the norm $\|\cdot\|_p$.
Hope everything is clear and thanks in advance for any help.
As I recalled, e.g., in
https://mathoverflow.net/questions/366983/known-dense-subset-of-schwartz-like-space-and-c-c-infty/368043#368043
The Hermite functions $h_n$ form an orthonormal basis of $L^2(\mathbb{R})$ and also an unconditional Schauder basis of Schwartz space $\mathscr{S}(\mathbb{R})$. It follows that the map from temperate distributions to sequences of at most polynomial growth $\mathscr{S}(\mathbb{R})\rightarrow \mathscr{s}'(\mathbb{N})$, $T\mapsto (T(h_n))_{n\in\mathbb{N}}$ is an isomorphism of topological vector spaces. The $h_n$ are eigenvectors for the Hamiltonian of the harmonic oscillator, which essentially is $A$. This is why $A$ is used in the norm. If you use the TVS isomorphism to toss the space $\mathscr{S}$ in the trash and instead work with $\mathscr{s}'$ throughout your white noise investigations, the norm will look rather simple and natural.