Lp Norm

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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: 2021-11-13

Emacs 26.1 (Org mode 9.5)