What makes the Cauchy-Schwarz inequality so important? The Cauchy-Schwarz inequality is $(a\cdot b)^2 \leq |a|^2|b|^2$.  Why is this considered such an important inequality: to quote my textbook it's "one of the most important inequalities in all of mathematics".  But why?  Doesn't it just immediately follow from the definition of the dot product: $a\cdot b = |a||b|\cos(\theta)$?  And even if you define the dot product differently, like maybe $a\cdot b = a_1b_1 + a_2b_2 +...$, it still doesn't seem all THAT important to me.  So what makes this particular inequality so important/ interesting?

The triangle inequality is an application of the Cauchy-Schwarz inequality. The triangle inequality is very important, especially since it is a condition for metric spaces. It is also very useful in probability theory with regards to the variance of $Y$ where $Y$ is a random variable. The Cauchy-Schwarz inequality also is important because it connects the notion of an inner product with the notion of length. 
