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.

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.

Those questions are rather hard to answer. An answer "The property $A$ is very important, because it can be used to prove properties $B$, $C$ and $D$ that are very important" does not clarify much but rise more questions why the latter are important. A definition of an "important result", that is the easiest to grasp by a non-professional, may come from a statistical estimation: if we count all known results (theorems etc) in analysis to be $N$, and let $M$ be the number of those that use a particular result, e.g. the Cauchy-Schwarz inequality, in their simplest proof (i.e. it would be much harder to prove without it), then $\frac{M}{N}$ may serve as a measure of importance. This kind of statistics is, of course, never calculated explicitly, but rather understood intuitively by an individuum who has a background in a certain subject, and others have to trust him until they accumulate their own statistical data.

Those who say that the Cauchy-Schwarz inequality is important may refer to that the ratio $\frac{M}{N}$ for this result is relatively high compared to the average. However, one can argue that the inequality $x^2\ge 0$, $\forall x\in\mathbb{R}$ is much more important in this sense.

An alternative (qualitative instead of quantitative) way to measure importance is perhaps to judge the influence of a particular notion and possibility to generalize it to other, more general constructions, that helps to study those. In this sense, the CS inequality is as important as the notion of a norm which was generalized from the plane and the $3D$ space to $\mathbb{R}^n$ first, and then to an abstract vector space, giving rise to Hilbert spaces, Banach spaces and a huge and very successful area of mathematics called functional analysis.