Norm

Definition

Let $\mathbb{K}$ be $\R$ or $\C$ , and let $V$ be a vector space over $\mathbb{K}$ . A norm on $V$ is a map $\norm{\cdot} \colon V \to \R$ such that, for all $x, y \in V$ and $a \in \mathbb{K}$ :

$\norm{x + y} \le \norm{x} + \norm{y}$ .
$\norm{ax} = \abs{a} \norm{x}$ .
If $\norm{x} = 0$ , then $x = 0$ .

Remarks

These conditions imply $\norm{x} \ge 0$ for all $x$ . On $\R^n$ , the Euclidean norm $\sqrt{x_1^2 + \cdots + x_n^2}$ , the norm $\abs{x_1} + \cdots + \abs{x_n}$ and the norm $\max_i \abs{x_i}$ are norms. Each inner product gives a norm, see the induced norm. A norm defines a metric $d(x, y) = \norm{x - y}$ .

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Vector space definition
Norm definition

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Tag	`08NYZ2FS6B`
URI	`m2://08NYZ2FS6B`
Kind	definition
Writing	W1 stated
MSC	46B20
Defines	Norm
Uses	Vector space

Norm

Definition

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