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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