Mathematics
The Cauchy-Schwarz Inequality and Its Many Faces
Quick fact
Cauchy-Schwarz is the mathematical reason why the correlation coefficient always lies between -1 and 1, a fact that every statistician relies on.
Why this is interesting
You might have noticed that the dot product of two vectors is never larger than the product of their lengths—no matter how long they are. But why is that true, and what does it have to do with statistics, geometry, and signals?