I’d often seen two different versions of Cauchy Schwartz (CS). In analysis and linear algebra I’d learned that if are elements of a vector space we have

(1)

while in probability and statistics I’d learned that for random variables we have

(2)

I’d seen proofs of both but not in a way that tied them together. Today, I learned how they’re related and it’s super simple. Define . One can verify that this satisfies the definition of an inner product. Then we have

(3)

it seems so obvious but I’d never realized this previously. Further, the statement

(4)

becomes really easy to prove using the inner product defined above.

(5)