Understanding the covariance of a measure

Question

I have found the following regarding the covariance operator on Wikipedia:

For a probability measure $\mathbb{P}$ on a Banach Space $B$, the covariance of $\mathbb{P}$ is the bilinear form on the dual space $B^\ast$ given by

$$\mathrm{Cov}(x,y) = \int_B x(z)y(z) \; \mathrm{d}\mathbb{P}(z), \quad \quad x,y \in B^\ast.$$

I believe that after reading this post I can understand the motivation for defining the covariance in this way. What I still struggle with is, why is the covariance a form on the dual space? And, this being the case, what does e.g. $\mathrm{Cov}(x,x)$ even tell us about the distribution?

Answer 1

Covariance measure is a generalization of covariance from probability theory, see e.g. https://en.wikipedia.org/wiki/Covariance

In $\S$ "Covariance of linear combinations" we see that $$ \rm{cov}(aX+bY, cW+dV) =ac * \rm{cov}(X,W)+ad * \rm{cov}(X,V)+bc * \rm{cov}(Y,W)+bd * \rm{cov}(Y,V) $$ so it's a bilinear form.

$\rm{cov}(X,X) = Var X$ is variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value.