Why is the Cauchy-Schwarz inequality considered to be so important?

Question

I've read in the book "Linear Algebra done right" by Axler that the Cauchy-Schwarz inequality is one of the most important results in mathematics. However, in what the book covers and what we have covered in our lecture, it's hardly used, except for proving the triangle inequality in an n-dimensional space. So I'm wondering what makes the inequality so important?

Answer 1

Well basically because you use it all the time, like the triangle inequality. It is one of these moves which you can always do, but there is nothing to it actually (the proof is very easy). For instance the Heisenberg inequality simply follows from C-S; also the C-S inequality is extremely general and holds for any inner product space.