# Lp Norm

The $$L^p$$ norm function is a type of Norm Functions. It is defined as follows:

For a given vector $$x \in \mathbb{R}^n$$ and $$0 < p < \infty$$,

$||x||_p \ = \left( \sum^n_{i=1} |x_i |^p \right)^{\frac{1}{p}}$

## 1.L1-Norm

$||x||_1 = \sum^n_{i=1} |x_i|$

## 2.L2-Norm

The $$L^2$$-Norm is also known as the Euclidean Norm.

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

## 3.L-Infinite-Norm

In this case, we must make use of the limit,

$||x||_{\infty} = \lim_{p \to \infty} \left(\sum^n_{i=1} |x_i|^p \right)^{\frac{1}{p}}$

This expression actually evaluates to the following! It is essentially equal to the max element of the vector $$x$$.

$||x||_{\infty} = \max_{1 \leq j \leq n} \left\{x_1, x_2, \ldots, x_n\right\}$

There is a detailed proof showing how to expand the limit, but no way I'm typing all that here.

Created: 2022-02-03

Emacs 26.1 (Org mode 9.5)