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Does this inequality have a name?

\begin{equation} \left| \sum_i x_i y_i \right| \leq \sum_i \left| x_i \right| \left| y_i \right| \end{equation}

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?

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

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It's the Minkowski or triangle inequality. –  Raskolnikov Feb 23 '12 at 13:42
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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
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up vote 2 down vote accepted

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

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