### Absolute-value norm

$$ | x| = |x| $$

is a norm on the $\Bbb R^1$ vector spaces formed by the real or complex numbers. Any norm $p$ on a $\Bbb R^1$ is equivalent (norm-preserving isomorphism of vector spaces).

### Euclidean norm

On the $\Bbb R^n$ Euclidean space, intuitive notion of length of the vector $\pmb x = (x_1, x_2, …, x_n)$ is

$$ |\pmb x|_2 := \sqrt{x^2_1+x^2_2+…+x^2_n} $$

or for two vectors

$$ |\pmb x|:= \sqrt{\pmb x\cdot \pmb x} $$

which is also called the $L^2$ norm, $\ell^2$ norm, 2-norm or square norm. And the distance defined by euclidean norm is called the Euclidean length, $L^2$ distance or $\ell^2$ distance.

### Euclidean norm of complex numbers

The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane $\Bbb R^2$ , the Euclidean norm is,

$$ |x + iy| = \sqrt{x^2 + y^2} $$

### Finite-dimensional complex normed spaces

On an n-dimensional complex space $\Bbb C^n$, the most common norm is

$$ |z||:=\sqrt{|z_1|^2 + … + |z_n|^2} = \sqrt{z_1\bar z_1 + … +z_n\bar z_n} $$

And can be expressed as the square root of the inner product of the vector and itself,

$$ |\pmb x|:=\sqrt{\pmb x^H\cdot \pmb x} $$

where, $\pmb x$ is represented as a column vector $[x_1;x_2;…;x_n]$, and $\pmb x^H$ denotes its conjugate transpose. Which also can be denoted with,

$$ |\pmb x|:=\sqrt{\pmb x\cdot \pmb x} $$

### Quaternions and octonions

There are exactly four Euclidean Hurwitz algebras over the real numbers. real number $\Bbb R$, the complex numbers $\Bbb C$, the quaternions $\Bbb H$ and the octonions $\Bbb O$, with dimension of 1, 2, 4 and 8, respectively.

### Taxicab norm or Manhattan norm

$$ |\pmb x|_1:= \sum_{i=1}^{n}|x_i| $$

Related to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point $x$ . Which can be also called the $\ell^1$ norm, and the distance derived from this norm is called the Manhattan distance or $\ell_1$ distance.

Attention, can’t written into, $\sum_{i=1}^{n}x_i$ instead, because it may yield negative results.

### p-norm

Let $p\ge1$ be a real number. The p-norm (also called $\ell_p$-norm) of vector $\pmb x = (x_1, …, x_n)$ is,

$$|\pmb x|p:=\left(\sum{i-i}^{n} |
x_i | ^p\right)^{1/p}$$ |

when $p = 1$, we get taxicab norm, $p=2$, we get Euclidean norm and if $p$ approaches $\infty$, we will get the infinity norm or maximum norm:

$$ |\pmb x|_{\infty}:=\max_i|x_i|. $$

The p-norm is related to the generalized mean or power mean.

when $0<p<1$ (wiki)

### Maximum norm (special case of : infinity norm , uniform norm or supremum norm)

$$ |x|_{\infty} :=max(|x_1|, …, |x+|) $$

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