Suppose $X_1, X_2, \cdots$ are iid standard Gaussians in $L_2$ for some $(\omega, \mathcal{F}, \mathbb{P})$. Let $H = \overline{span\{X_1, X_2, \cdots\}}$ (i.e the closure) in $L_2$.
We want that for every $p > 2$, $H$ is closed in $L_p$ and that there is a linear operator $P:L_p \rightarrow H$ such that $P^2 = P$ (ie a projection) and $||PY||_p \leq C||Y||_p$ for some constant $C \in \mathbb{R}^+$ and every $Y \in L_p$ (so $P$ is continuous).
I would appreciate some hints. The $L_p$ space stuff is completely unintuitive to me. I was thinking of using the result that for every $p$, and every complete $A \subseteq L_p$, and for every $X \in L_p$ there is a nearest point $Y \in A$ such that $||X-Y||_p = ||X-A||_p$ and $X-Y$ is orthogonal the space. But I don't think it works here. I also don't understand why $p>2$ is necessary. I'm not sure what the natural approximation of a RV by Gaussians would be. In summation, I would like some help.