I get that a zero covariance doesn´t imply independence, but everybody says that if there is dependence and the covariance is zero then it is a non linear dependence.
People base their interpretation of Pearson's R in that fact (the closer you are to zero the less linear the relationship is).
Is there a formal proof to that?
I tried to do it by myself but i couldn't. The proposition i think encapsulates the idea is the following:
If $cov(X,Y)\ne0$ then there exists a Z such that $cov(X,Z)=0$ and $E[Y|X]=bX+E[Z|X]$