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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.2012-02-23
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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 $w$?2012-02-23
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    @Raskolnikov Isn't the triangle inequality $|\sum_i x_i| \leq \sum_i |x_i|$?2012-02-23
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    @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 $|x_i| |y_i|$?2012-02-23
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    Yes, it is. See Michael's comment.2012-02-23
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    @Raskolnikov So is it still a triangle inequality? And is the actual inequality then $|\sum_i w_i| \leq \sum_i |w_i| \leq \sum_i |x_i||y_i|$?2012-02-23
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    It is the triangle inequality with $w_i=x_i y_i$.2012-02-23
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    @Raskolnikov But that would leave you with $|\sum_i w_i| \leq \sum_i |x_i y_i|$ ...not the modulus of each of $|x_i|$ and $y_i$.2012-02-23
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    @Calvin But $|x_iy_i|=|x_i||y_i|$...2012-02-23
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    @AlexBecker Excellent point... I somehow missed that!2012-02-23

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