A simple proof of Cauchy-Schwarz inequality

18Sep07

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:

<x,y> \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

<\frac{x}{\left\|x \right\|}-\frac{y}{\left\|y \right\|},\frac{x}{\left\|x \right\|}-\frac{y}{\left\|y \right\|}> \geq 0

<\frac{x}{\left\|x \right\|},\frac{x}{\left\|x \right\|}> + <\frac{y}{\left\|y \right\|},\frac{y}{\left\|y \right\|}> -2<\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}> \geq 0

1+1 -2<\frac{x}{\left\|x \right\|},\frac{y}{\left\|y \right\|}> \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.

Filed under: Basics   |  

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