[reader be aware: this is a post for getting attention from the web, you may get these information elsewhere and probably also better written]

There are a lot of proof about Cauchy-Schwarz inequality, there is also an entire book on it.

This is a proof without many advantages with respect to others, rather being very mechanic, however it is the only way someone reached my blog 🙂

Cauchy-Schwarz inequality states that:

\langle x,y\rangle \leq \left\|x \right\| \left\|y \right\|

Let’ s see that this inequality comes shortly from axioms of a vector space and from definition of scalar product

\langle\frac{x}{\left\|x \right\|}-\frac{y}{\left\|y \right\|}, \frac{x}{\left\|x \right\|}-\frac{y}{\left\|y\right\|} \rangle \geq 0

\langle\frac{x}{\left\|x \right\|},\frac{x}{\left\|x \right\|}\rangle + \langle\frac{y}{\left\|y \right\|},\frac{y}{\left\|y \right\|}\rangle -2\langle\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}\rangle \geq 0

1+1 -2\langle\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}\rangle \geq 0

and inequality follows.

The quality of this proof is that it comes out from the axioms of a vector space, unfortunately it does not bring an intuitive insight about this fundamental tool for analysis.