Norm
Concept history
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Revision 3007
8/30/2026, 6:12:07 PM · La chouette aveugle
Added exercise "Irrégularité des bases algébriques dans un espace de Banach"
linked exercisesExemples de normes, Existence de norme, Espaces vectoriel normés quotient et semi-normesExemples de normes, Existence de norme, Espaces vectoriel normés quotient et semi-normes, Irrégularité des bases algébriques dans un espace de Banach
Revision 2996
8/30/2026, 5:38:59 PM · La chouette aveugle
Added exercise "Espaces vectoriel normés quotient et semi-normes"
linked exercisesExemples de normes, Existence de normeExemples de normes, Existence de norme, Espaces vectoriel normés quotient et semi-normes
Revision 2977
8/30/2026, 5:08:07 PM · La chouette aveugle
Added exercise "Existence de norme"
linked exercisesExemples de normesExemples de normes, Existence de norme
Revision 2974
8/30/2026, 5:06:26 PM · La chouette aveugle
Added exercise "Exemples de normes"
linked exercisesNoneExemples de normes
Revision 1439
8/16/2026, 8:43:18 AM · Ancient Tree
Updated text
Compare with revision 14363 changed lines
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##### Intuition2
A norm is an object which can measure the "size" or "length" of a vector.3
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##### Formal definition5
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A norm on a [[Vector space|vector space]] $E$ is a function $N:E\rightarrow \R^{+}$ with the following properties, for all vectors $x,y\in E$ and scalars $a\in \R$:7
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1. Separation: If $N(x)=0$, then $x=0$.9
2. Scaling: $N(ax)=|a|N(x)$.10
3. Triangle inequality: $N(x+y)\leq N(x)+N(y)$.11
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##### Remarks13
- Norms are usually denoted14
$$\|.\|$$15
For instance, $\|(x,y)\|$ is the norm of vector $(x,y)$ in $\R^{2}$.13
- A vector space with a norm is called a [[Normed vector space|normed vector space]].Revision 1436
8/16/2026, 8:37:08 AM · Ancient Tree
Concept created
##### Intuition
A norm is an object which can measure the "size" or "length" of a vector.
##### Formal definition
A norm on a [[Vector space|vector space]] $E$ is a function $N:E\rightarrow \R^{+}$ with the following properties, for all vectors $x,y\in E$ and scalars $a\in \R$:
1. Separation: If $N(x)=0$, then $x=0$.
2. Scaling: $N(ax)=|a|N(x)$.
3. Triangle inequality: $N(x+y)\leq N(x)+N(y)$.
##### Remarks
- A vector space with a norm is called a [[Normed vector space|normed vector space]].