Well-defined sum on Schauder-basis of Hilbert space

Question

I am struggling with the follwing question:
Let $(X_k)_{k \in \mathbb{N}}$ be a sequence of i.i.d. standard normally distributed random variables on $(\Omega, \sigma, \mathbb{P})$.
Let $H := L^2(\mathbb{R}_+, dx)$ with $\langle f,g \rangle_H = \int_0^{\infty} f(x)g(x) dx$ be a separable Hilbert space with orthonormal Schauder-basis $(b_n)_{n \in \mathbb{N}}$, that means, that for every $h \in H $ it is:
$\lim _{{n\to \infty }}\left\|h-\sum _{{k=0}}^{n}\langle h,b_k \rangle b_{k}\right\|_{L^2(\mathbb{P})}=0$

I want to show, that $\| \sum_{k \in \mathbb{N}} \langle h, b_k \rangle_H X_k \|_{L^2(\mathbb{P})}^2 < \infty$

So far I found out, that $\mathbb{E}(X_k) = 0$ and $\| X_k\|_{L^2(\mathbb{P})}^2 = \mathbb{E}(X_k^2) = \int_\Omega X_k^2 d\mathbb{P} = \operatorname{Var}(X) = 1$ for all $k \in \mathbb{N}$, since $X_k \sim \mathcal{N}(0,1)$.
Also I tried to solve it by using the triangle inequality and homogeneity:

$\| \sum_{k \in \mathbb{N}} \langle h, b_k \rangle_H X_k \|_{L^2(\mathbb{P})}^2 \leq \sum_{k \in \mathbb{N}} \| \langle h, b_k \rangle_H X_k \|_{L^2(\mathbb{P})}^2 \leq \sum_{k \in \mathbb{N}} |\langle h, b_k \rangle_H|^2 \cdot \| X_k \|_{L^2(\mathbb{P})}^2 \leq \sum_{k \in \mathbb{N}} |\langle h, b_k \rangle_H|^2$

I don't know, if this is the right attempt, since I can't think of an argument, why this last sum would be finite.
Certainly, without the absolute value $\sum_{k \in \mathbb{N}} \langle h, h \rangle_H = \| h \|_H$ would converge. I would appreciate, if one of you has a good idea, how to solve this, thank you.

Answer 1

Recall Bessel's inequality (https://en.wikipedia.org/wiki/Bessel%27s_inequality)

Indeed, this says that $\sum_k |\langle h, b_k \rangle |^2 \leq \Vert h\Vert^2$

Thus

$$\sum_k \Vert\langle h, b_k \rangle X_k \Vert^2=\sum_k |\langle h, b_k \rangle |^2 \Vert X_k \Vert^2 = \sum_k |\langle h, b_j \rangle |^2 < \infty$$

as desired.

Answer 2

You have a typo in the right-hand side of your second inequality. It should be $$ \sum_{k\in\mathbb{N}}\left\|\left\langle h,b_k\right\rangle_HX_k\right\|_{L^2(\mathbb{P})}^2 \le \sum_{k\in\mathbb{N}} \left|\left\langle h,b_k\right\rangle_H\right|^2\left\|X_k\right\|_{L^2(\mathbb{P})}^2\ , $$ and since $\ \displaystyle \sum_{k\in\mathbb{N}} \left|\left\langle h,b_k\right\rangle_H\right|^2=\|h\|_H^2\ $, this gives you your result.