Proving Schwarz inequaltiy - how to begin

[spaghetti3451]

Well, I have solved this problem from a textbook:

Prove the Schwarz inequality.

When you try to solve it, you wonder - how do i begin? But thankfully, the problem had a hint: Let $\left|γ\right\rangle = \left|β\right\rangle - \frac{\left\langleα\right| \left|β\right\rangle}{\left\langleα\right| \left|α\right\rangle} \left|α\right\rangle$, and use $\left\langleγ\right| \left|γ\right\rangle \geq 0$.

Well, with this hint, the problem becomes a piece of cake. But how would you know where to start if you didn't have the hint? That's the problem - the starting point. Any ideas?

 

[dodo]

(The self-admitted voice of ignorance speaking here.) I'm not familiar with bra-ket notation, but see for yourself if this hint is useful to you.

These are vectors in some space, and the underlying subject trying to bubble up here is "how to define what an angle between vectors is". The Cauchy-Schwarz inequality boils down, in simpler contexts, to stating that the cosine of such angle is less than or equal to one. Maybe you can use this (even if just as an analogy) to guide your aim.