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 Cauchy-Schwarz inequality holds for much wider range of settings than just the two- or three-dimensional Euclidean space $\mathbb R^2$ or $\mathbb R^3$.
In fact, it holds for all kind of spaces, where an inner product (an abstract concept) is defined. Thus it can be applied to bound things in wide number of settigns.
