I mentioned that there is a 318 page book completely devoted to the Cauchy-Schwarz inequality. It is The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Its author, Michael J. Steele, is a past President of the Institute of Mathematical Statistics. One reviewer wrote "... this is one of that handful of mathematics books that you can read almost like a novel." Chapter 1 is free to download and really interesting to read. I can also recommend his highly entertaining web site, which includes material for advanced courses in probability and finance, and also Semi-Random Rants, Videos and Favorite Quotes.
There are many proofs of this inequality. Here are two in addition to the one I did in lectures.
1. Consider $E(tX-Y)^2]=t^2E[X^2]-2t E[XY]+E[Y^2]\geq 0$, and $>0$ unless $tX=Y$. This quadratic has most one real roots and so the discriminant must be nonpositive. This is essentially the proof that was given by Schwarz and is a textbook favorite.
2. We started the lecture with Jensen's inequality. It is natural to ask if the Cauchy-Schwarz inequality can be deduced from it. One version of the Cauchy-Schwarz inequality is
$(\sum_i a_ib_i)^2\leq \sum_ia_i^2\sum_ib_i^2$,
for all numbers $(a_1,\dotsc,a_n)$ and $(b_1,\dotsc,b_n)$. WLOG we may assume all $a_j,b_j> 0$. Consider a random variable $X$ for which $P(X=a_j/b_j)=b_j/\sum_ib_i^2$. Apply Jensen's inequality with the choice of convex function $f(x)=x^2$. $Ef(X)\geq f(EX)$ gives the result.
This paper presents 12 different proofs of the Cauchy-Schwarz inequality, and this web page has 10.
There are many proofs of this inequality. Here are two in addition to the one I did in lectures.
1. Consider $E(tX-Y)^2]=t^2E[X^2]-2t E[XY]+E[Y^2]\geq 0$, and $>0$ unless $tX=Y$. This quadratic has most one real roots and so the discriminant must be nonpositive. This is essentially the proof that was given by Schwarz and is a textbook favorite.
2. We started the lecture with Jensen's inequality. It is natural to ask if the Cauchy-Schwarz inequality can be deduced from it. One version of the Cauchy-Schwarz inequality is
$(\sum_i a_ib_i)^2\leq \sum_ia_i^2\sum_ib_i^2$,
for all numbers $(a_1,\dotsc,a_n)$ and $(b_1,\dotsc,b_n)$. WLOG we may assume all $a_j,b_j> 0$. Consider a random variable $X$ for which $P(X=a_j/b_j)=b_j/\sum_ib_i^2$. Apply Jensen's inequality with the choice of convex function $f(x)=x^2$. $Ef(X)\geq f(EX)$ gives the result.
This paper presents 12 different proofs of the Cauchy-Schwarz inequality, and this web page has 10.
