In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.
Today’s quip is one that I’ll use surprisingly often:
If you ever meet a mathematician at a bar, ask him or her, “What is your favorite application of the Cauchy-Schwartz inequality?”
The point is that the Cauchy-Schwartz inequality arises surprisingly often in the undergraduate mathematics curriculum, and so I make a point to highlight it when I use it. For example, off the top of my head:
1. In trigonometry, the Cauchy-Schwartz inequality states that
for all vectors and . Consequently,
which means that the angle
is defined. This is the measure of the angle between the two vectors and .
2. In probability and statistics, the standard deviation of a random variable is defined as
The Cauchy-Schwartz inequality assures that the quantity under the square root is nonnegative, so that the standard deviation is actually defined. Also, the Cauchy-Schwartz inequality can be used to show that implies that is a constant almost surely.
3. Also in probability and statistics, the correlation between two random variables and must satisfy
Furthermore, if , then for some constants and , where . On the other hand, if , if , then for some constants and , where .
Since I’m a mathematician, I guess my favorite application of the Cauchy-Schwartz inequality appears in my first professional article, where the inequality was used to confirm some new bounds that I derived with my graduate adviser.