Projecting bounded functions on a Gaussian Hilbert Space Let $\{g_n\}_{n=1}^{\infty}$ be an infinite sequence of mutually independent, normally distributed random variables, all with zero expectation and unit-variance, all defined in some probability space. I believe I can prove that if $f$ is a real-valued bounded function, then
$$\sum_{n=1}^{\infty}[\mathbf{E}(fg_n)]^2\leq\frac{2}{\pi}\sup|f|^2$$
and that $\frac{2}{\pi}$ is the best possible constant, attained by certain bounded functions.

My question is this: has anyone seen this before and can provide reference?

 A: As noted by the OP in a comment, the inequality
$$\sum_{n=1}^{\infty}[\mathbf{E}(fg_n)]^2\leq\frac{2}{\pi}\sup|f|^2 \quad(*)$$
is true if $\{g_n\}$ is replaced by  a single standard Gaussian variable $g$, because $$[E(fg)]^2 \le (\sup |f|)^2 [E|g|]^2 =\frac{2}{\pi} (\sup |f|)^2  \,. \quad (**)$$
The inequality $(**)$ is tight because you can take $f={\rm sgn}(g)$.
The general case $(*)$ may be reduced to the special case above by considering the
projection
$$Z:=\sum_{n=1}^\infty E(fg_n)g_n$$
and its $L^2$-normalized version, $g=Z/\|Z\|_2$.
Observe that $f-Z$ is orthogonal to every $g_n$, so $E[(f-Z)g]=0$.
Then
$$\sum_{n=1}^{\infty}[\mathbf{E}(fg_n)]^2 =E(Z^2)=[E(Zg)]^2=[E(fg)]^2$$
and we are done by $(**)$.
Edit:  As noted in the comment by Nate Eldredge, an orthonormal basis for $L^2(\Omega,{\cal F},P)$ cannot be entirely comprised of independent centered Gaussians (together with 1), since taking $f={\rm sgn (g_1)}$ in the above inequality would contradict Parseval's identity. You can see this even more easily by noting that, given a  sequence of independent centered gaussians $\{g_n\}$, the random variable $g_1^2-1$ is orthogonal to $g_n$ for all $n \ge 1$ and to the constant functions. (The same is true for the random variable $g_1g_2$.) Taking this further leads to Hermite polynomials [1] and Hermite polynomial chaos [2].
[1]  https://en.wikipedia.org/wiki/Hermite_polynomials
[2] https://en.wikipedia.org/wiki/Polynomial_chaos
