The relationship between covariance and inner product It looks like covariance has some relationship between inner product. I have checked that covariance is bilinear and symmetric, but not positive definite (it is positive semi-definite). Although in WIKI, there are some information about the relationship between covariance and inner product, I still cannot really get the point. Hope someone will give me a more detailed answer about the relationship.

If $X$ and $Y$ are random variables with finite second moments on a probability space $(\Omega, \mathcal F ,P)$  and $L^{2}$ is the space of square integrable random variables on $(\Omega, \mathcal F ,P)$ with the inner product $ \langle X, Y \rangle=EXY$ then covariance of $X$ and $Y$ is the inner product of $X-EX$ and $Y-EY$.
