Euclidean norm

This is an example of a Norm

Definition

The Euclidean norm is denoted $\|\cdot\|_2$ is a norm on $\mathbb{R}^n$

Here for $x\in\mathbb{R}^n$ we have:

$\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}$

Proof that it is a norm

TODO: proof

Part 4 - Triangle inequality

Let $x,y\in\mathbb{R}^n$

$\|x+y\|_2^2=\sum^n_{i=1}(x_i+y_i)^2$ $=\sum^n_{i=1}x_i^2+2\sum^n_{i=1}x_iy_i+\sum^n_{i=1}y_i^2$ $\le\sum^n_{i=1}x_i^2+2\sqrt{\sum^n_{i=1}x_i^2}\sqrt{\sum^n_{i=1}y_i^2}+\sum^n_{i=1}y_i^2$ using the Cauchy-Schwarz inequality

$=\left(\sqrt{\sum^n_{i=1}x_i^2}+\sqrt{\sum^n_{i=1}y_i^2}\right)^2$ $=\left(\|x\|_2+\|y\|_2\right)^2$

Thus we see: $\|x+y\|_2^2\le\left(\|x\|_2+\|y\|_2\right)^2$, as norms are always $\ge 0$ we see:

$\|x+y\|_2\le\|x\|_2+\|y\|_2$ - as required.