This tag is for the questions relating to the norm which is a function on a vector space $X$ that generalizes notion of length of vector in general vector spaces. A vector space $~X~$ with a distinguished norm is called a normed space.

The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object.

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero.

Norms exist for complex numbers (the complex modulus, sometimes also called the complex norm or simply "the norm"), Gaussian integers (the same as the complex modulus, but sometimes unfortunately instead defined to be the absolute square), quaternions (quaternion norm), vectors (vector norms), and matrices (matrix norms).

Definition: A norm on a vector space $V$ over the field $~{K}~$ (that is, $\mathbb R$ or $\mathbb C$) is a function $\Vert\cdot\Vert\colon V\longrightarrow\mathbb{R}_+$ such that

$ 1.\quad \Vert\alpha~ v\Vert=|\alpha|~\Vert v\Vert\quad \forall~ v\in V~,~\forall~\alpha\in~{K}~$

$ 2.\quad \Vert v+w\Vert\leq\Vert v\Vert+\Vert w\Vert\quad \forall~ v,w\in V$

$ 3.\quad \Vert v\Vert\geq 0\qquad \text{for any $~v\in ~V~$}$

$4.\quad \Vert v\Vert =0\Longleftrightarrow v=0\quad \forall ~v\in V $

For example the function $\Vert (x_1,\ldots,x_n)\Vert=\sqrt{\sum\limits_{k=1}^n|x_k|^2}$ defines a norm on the complex vector space $\mathbb{C}^n$.


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