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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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    @AlexBe$c$ker 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 $|ab|=|a|\,|b|$.