I am currently reading a book "measure, integral and probability" by Capinski and Kopp. The correlation between random variables $X$ and $Y$ is defined as the cosine of the angle between $X_c$ and $Y_c$, that is: $$ \operatorname{corr(X,Y)} = \frac{(X_c,Y_c)}{(\|X\|\cdot\|Y\|)}, $$ where $X_c$, $Y_c$ are centered random variables defined by $X_c=X-\mathbb{E}(X)$, $Y_c=Y-\mathbb{E}(Y)$. The question that immediately arises is: Why do we divide by $\|X\|\cdot\|Y\|$, and not $\|X_c\|\cdot\|Y_c\|$?
I want to understand the concept of correlation, independence and other concepts of probability from the point of view of functional analysis. I would be very happy if you could recommend some literature that has a treatment of this topic, and desirably, detailed discussion of the following questions: Are uncorrelated variables just orthogonal? or only if $\mathbb{E}(X)=0=\mathbb{E}(Y)$? How is the "centred" vector different from original, "geometrically"? I know in infinite dimenstions it is hard to visualise geometry, but still... Thank you very much.