# Cauchy-Schwarz sanity check

Cauchy Schwarz say that $$\mid x_1y_1\mid +...+ \mid x_ny_n\mid \leq \sqrt{x_1^2+...+x_n^2}\sqrt{y_1^2+...+y_n^2}.$$ (This follows from the popular proof using AM-GM.)

This is Holder's inequality with conjugate pairs (2,2) if you like. But it seems like everyone writes it as $$\mid x_1y_1+...+ x_ny_n\mid \leq \sqrt{x_1^2+...+x_n^2}\sqrt{y_1^2+...+y_n^2}.$$ The second one follows from the first one but the first one does not follow from the second, so why is the second way of writing it vastly more popular?

• I have never seen CS stated in the first manner. Also, the first does follow from the second. – copper.hat May 8 '17 at 22:01
• I agree with @copper.hat – Chee Han May 8 '17 at 22:01
• Actually these two formulations are equivalent. Simply apply the second inequality to the vectors $(|x_1|, \dots , |x_n|)$ and $(|y_1|, \dots , |y_n|)$. – Crostul May 8 '17 at 22:02
• I've seen it formulated the first way. It is seemingly stronger, but as Crostul points out the second implies the first. – Robert Wolfe May 8 '17 at 22:15
• The 2nd is also a direct fit to its generalization to inner products on vector spaces $|u\cdot v| \le \|u\|\|v\|$, so it is not surprising to me that this form would be preferred. – Paul Sinclair May 9 '17 at 2:55