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Let $x_i, y_i$ be complex numbers for all $i$.

Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$

In particular, is it a special case of this form of the Cauchy-Schwarz Inequality?

$$\left| \sum_{i=1}^n x_i \bar{y}_i \right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 $$

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    $\begingroup$ This is the triangle inequality. $\endgroup$ – user1337 Jan 25 '15 at 0:56
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    $\begingroup$ This inequality is often a defining property for norms/distance measures. In such cases Cauchy Schwarz inequality itself relies on this property. Where the norm is defined by an inner product, Cauchy Schwarz can be used to prove the triangle inequality. $\endgroup$ – Macavity Jan 25 '15 at 2:18
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    $\begingroup$ In inner products spaces you can get triangle inequality from Cauchy-Schwarz. You should be able to find relatively easily such proofs online. $\endgroup$ – Martin Sleziak Jan 25 '15 at 11:01
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Usually it's referred to as the triangle inequality.

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  • $\begingroup$ Ah thanks! Can it be derived in a simple way from the Cauchy-Schwarz, or is it a different animal altogether? $\endgroup$ – TSJ Jan 25 '15 at 0:58
  • $\begingroup$ @user208884 it is the same as the triangle inequality for $R^2$, which in fact can be derived from C-S. $\endgroup$ – quid Jan 25 '15 at 0:59
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    $\begingroup$ @user208884 It can indeed be derived from Cauchy Schwarz. See here. $\endgroup$ – Peter Woolfitt Jan 25 '15 at 1:00

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