I have been given a couple of properties of inner product and a norm in these cambridge notes on metric and topological spaces, page 6.
And then lemma 3.5 (i) talks about cauchy-schwarz inequality. I have looked at their proof and they show that:
$<u,u>-\frac {<u,v>^2}{<v,v>} \geq 0$ and this should be equivalent to $||u||||v|| \geq |<u,v>|$. I fail to see it. Especially, given that they have not shown an explicit form of either the norm or an inner product, rather their properties.
You have noticed something very important:
This is a level of abstraction and allows flexible adaptation of the inner product. The results shown for such inner products are thus applicable to all mappings satisfying the inner product axioms.
By definition, $\|u\|^2 = \langle u,u \rangle$. Hence $$ \langle u,u \rangle - \frac{\langle u,v \rangle^2}{\langle v,v \rangle} \geq 0 \implies \langle u,u \rangle \geq \frac{\langle u,v \rangle^2}{\langle v,v \rangle} \implies \langle u,u \rangle \langle v,v\rangle \geq \langle u,v\rangle^2 \implies \|u\|^2 \|v\|^2 \geq \langle u,v\rangle^2 $$