Norm - Revision history

Rongzhou at 07:59, 1 October 2026

2026-10-01T07:59:19Z

← Older revision		Revision as of 07:59, 1 October 2026
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	* '''Homogeneity''': $\forall a\in\mathbb{F},\;\forall v\in V,\; \left\\| av \right\\|=\|a\|\left\\| v \right\\|$;		* '''Homogeneity''': $\forall a\in\mathbb{F},\;\forall v\in V,\; \left\\| av \right\\|=\|a\|\left\\| v \right\\|$;
	* '''Triangle inequality''': $\forall v,w\in V,\; \left\\| v+w \right\\|\le \left\\| v \right\\|+\left\\| w \right\\|$		* '''Triangle inequality''': $\forall v,w\in V,\; \left\\| v+w \right\\|\le \left\\| v \right\\|+\left\\| w \right\\|$
			<ref>{{#cite:Q1920}}</ref>

	<blockquote>		<blockquote>

2026-09-30T21:06:16Z

← Older revision		Revision as of 21:06, 30 September 2026
(No difference)

2026-09-30T21:05:30Z

Positive definiteness

2026-09-30T21:05:04Z

← Older revision		Revision as of 21:05, 30 September 2026
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	~~== Overview ==~~		In mathematics, a norm is a function in [[vector space\|vector spaces]] with certain special properties.

	mathematical ~~operation~~		== Conventional definition in inner product spaces ==

			By convention, in [[inner product space]] $V$, the norm of a vector $v \in V$, denoted $\left\\| v \right\\|$, is defined:

			$$
			\\| v \\| = \sqrt{\langle v, v \rangle}
			$$

			== Formal definition in all vector spaces ==

			Let $V$ be a vector space over $\mathbb{F}$.

			Map

			$$
			\begin{aligned}
			\\| \cdot \\|: V &\to \mathbb{R}\\
			v &\mapsto \\| v \\|
			\end{aligned}
			$$

			is a '''norm''' on $V$ if 3 conditions are satisfied:

			* '''Definiteness''': $\forall v\in V,\; \left\\| v \right\\|=0 \iff v=\vec 0$;
			* '''Homogeneity''': $\forall a\in\mathbb{F},\;\forall v\in V,\; \left\\| av \right\\|=\|a\|\left\\| v \right\\|$;
			* '''Triangle inequality''': $\forall v,w\in V,\; \left\\| v+w \right\\|\le \left\\| v \right\\|+\left\\| w \right\\|$

			<blockquote>
			Notice the conventional definition $\\| v \\| = \sqrt{\langle v, v \rangle}$ is a special case of the mathematical definition.
			</blockquote>

			== Derived properties ==

			=== Positive definiteness ===

			$$\forall v \in V \\| v \\|\ge 0$$

			'''Proof:'''

			$$
			\\| v \\| = \frac{1}{2}(2\\| v \\|)= \frac{1}{2}(\\| v \\| + \\| -v \\|) \\
			\stackrel{\text{Triangle inequality}}{\ge} \frac{1}{2}\\| v-v \\| = 0
			$$

2026-09-29T12:58:28Z

Created the wiki page for norm (vector space operation)

New page

== Overview ==

mathematical operation