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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?

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    $\begingroup$ I have never seen CS stated in the first manner. Also, the first does follow from the second. $\endgroup$ – copper.hat May 8 '17 at 22:01
  • $\begingroup$ I agree with @copper.hat $\endgroup$ – Chee Han May 8 '17 at 22:01
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    $\begingroup$ Actually these two formulations are equivalent. Simply apply the second inequality to the vectors $(|x_1|, \dots , |x_n|)$ and $(|y_1|, \dots , |y_n|)$. $\endgroup$ – Crostul May 8 '17 at 22:02
  • $\begingroup$ I've seen it formulated the first way. It is seemingly stronger, but as Crostul points out the second implies the first. $\endgroup$ – Robert Wolfe May 8 '17 at 22:15
  • $\begingroup$ 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. $\endgroup$ – Paul Sinclair May 9 '17 at 2:55

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