Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $X_1,\cdots,X_n:\Omega\to\mathbb{R}$ be independent normally distributed random variables. Show that if $M_1,\cdots,M_p,N_1,\cdots,N_q\in\text{Span}(X_1,\cdots,X_n)$ and $\text{Cov}(M_i,N_j)=0$ for all $i,j$, then $(M_1,\cdots,M_p)$ and $(N_1,\cdots,N_q)$ are independent.
There is a proof of the above proposition in the book "Mathematical Statistics" written by Wiebe R. Pestman (see the section 1.6).
That proof uses the notion of orthonormal basis with respect to the covariance, however, since covariance isn't a inner product, I'd like to know is there's a proof of the above proposition without using such notion.
Thank you for your attention!
The vector $V=(M_1,\dots,M_p,N_1,\dots,N_q)$ is Gaussian because any linear combination of $V$ can be expressed as a linear combination of the vector $(X_1,\dots,X_n)$, which is Gaussian. By assumption, the covariance matrix of $V$ is a block matrix hence the density of $V$ is the product of a density putting into play $x_1,\dots,x_p$ and an other one $x_{p+1},\dots,x_{q}$.