# Does this inequality have a name? - Upper bound of the modulus of a sum

Does this inequality have a name?

$$\left| \sum_i x_i y_i \right| \leq \sum_i \left| x_i \right| \left| y_i \right|$$

If not (which means searching for information on it will be difficult), is it true for complex numbers as well as real ones? And does the same inequality apply to integrals?

$$\left| \int dt \hspace{1mm} x(t) y(t) \right| \leq \int dt \hspace{1mm} \left| x(t) \right| \left| y(t) \right|$$

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It's the Minkowski or triangle inequality. – Raskolnikov Feb 23 '12 at 13:42
It's not clear why you're multiplying two things instead of just working with one thing, which is the product. Let $w_i=x_i y_i$; then the inequality says $\left| \sum_i w_i \right| \le \sum_i |w_i|$. What is gained by having a factorization of $w$? – Michael Hardy Feb 23 '12 at 13:49
@Raskolnikov Isn't the triangle inequality $|\sum_i x_i| \leq \sum_i |x_i|$? – Calvin Feb 23 '12 at 14:00
@MichaelHardy Because I have something where I can't calculate the sum $\sum_i |w_i|$ with $w_i = x_i y_i$, but I do know that $|y_i| \leq K$ where $K$ is a constant. ...Is it still a type of triangle inequality if I need it in terms of $|x_i| |y_i|$? – Calvin Feb 23 '12 at 14:08
Yes, it is. See Michael's comment. – Raskolnikov Feb 23 '12 at 14:08

Just so we have an answer - it's the triangle inequality, together with the observation that $|ab|=|a|\,|b|$.