 Perfect positive correlation

For square-integrable variables of positive variance, correlation one is equivalent to $X-\mathbb{E}X=c(Y-\mathbb{E}Y)$ with $c>0$. The squared norm of the difference between the two standardized centered variables is $2(1-\rho )$, proving the equivalence. For uncentered variables the relationship can have a nonzero intercept; it need not be proportionality.