given a vector space (𝑉,𝐹), a norm/semi-norm (||·||) is a real-valued [[Continuous／Continuity [at a point - everywhere] (Transformations／Operations／Operations／Mappings／Maps／Functions／Morphisms)|continuous function]] on 𝑉 that takes in a vector from 𝑉 and outputs a non-negative real number (i.e. ||·||: 𝑉 → ℝ⁺)
a vector space with a norm is called a normed vector space
used to measure the length of a vector
an inner product (⟨·,·⟩) induces a norm (||·||_⟨·,·⟩) defined as: ||·||_⟨·,·⟩ = √⟨·,·⟩
a norm (||·||)induces a distance metric (𝑑||·||) defined as: 𝑑_||·||(𝑥,𝑦) = ||𝑥-𝑦||

Norm - Semi-Norm - Definitions

	Properties
	Triangle Inequality for all 𝑣,𝑢∈𝑉 Also called Subadditivity \|\|𝑣+𝑢\|\| ≤ \|\|𝑣\|\| + \|\|𝑢\|\|	Absolute Homogeneity for all 𝑣∈𝑉and all scalars 𝑠 \|\|𝑠𝑣\|\| = \|𝑠\|·\|\|𝑣\|\|	Non Negativity for all 𝑣∈𝑉 Some authors include non-negativity as part of the definition of “norm”, although this is not necessary. Although this article defined “positive” to be a synonym of “positive definite”, some authors instead define “positive” to be a synonym of “non-negative”; these definitions are not equivalent. \|\|𝑣\|\| ≥ 0	Positive Definiteness Also called Positiveness or Point-Separating Because property (2) implies \|\|0\|\| = 0, some authors replace the property (3) with the equivalent condition: for every 𝑣∈𝑉, \|\|𝑣\|\| = 0 if and only if 𝑣 = 0 \|\|𝑣\|\| = 0 ↔ 𝑣 = 0
Norm	REQUIRED	REQUIRED	IMPLIED	REQUIRED
Semi-Norm	REQUIRED	REQUIRED	IMPLIED

Types of Vector Norms

Type	Category	Input	Description	Relationships Between Lp Vector Norms
L0 “Norm”	FAILS HOMOGENEITY	vector	𝐿₀ = \|\|𝑣̅\|\|₀ = 𝛴_{1≤𝑖≤𝑛}\|𝑣̅_𝑖\|⁰(where 0⁰ = 0)
Absolute Norm	NORM	vector	𝐿₁ = \|\|𝑣̅\|\|₁ = 𝛴_{1≤𝑖≤𝑛}\|𝑣̅_𝑖\|
Euclidean Norm	NORM	vector	𝐿₂ = \|\|𝑣̅\|\|₂ = \|\|𝑣̅\|\| = [ 𝛴_{1≤𝑖≤𝑛}(𝑣̅_𝑖)² ]^(1/2)	\|\|𝑣̅\|\|₂ ≤ \|\|𝑣̅\|\|₁
Minkowski Norm	NORM	vector	𝐿_𝑝 = \|\|𝑣̅\|\|_𝑝 = [ 𝛴_{1≤𝑖≤𝑛}\|𝑣̅_𝑖\|^𝑝 ]^(1/𝑝)	\|\|𝑣̅\|\|_𝑝+1 ≤ \|\|𝑣̅\|\|_𝑝
Tchebychev Norm	NORM	vector	𝐿_∞ = \|\|𝑣̅\|\|_∞ = [ 𝛴_{1≤𝑖≤𝑛}\|𝑣̅_𝑖\|^∞ ]^(1/∞) = 𝑚𝑎𝑥(\|𝑣̅_𝑖\|)	\|\|𝑣̅\|\|_∞ ≤ \|\|𝑣̅\|\|_𝑝

Type	Category	Input	Description
Matrix Lpq Norm	NORM	matrices	𝐿_𝑝𝑞 = \|\|𝐴\|\|_𝑝𝑞= (𝛴_𝑗(𝛴_𝑖\|𝐴_𝑖𝑗\|^𝑝)^(𝑞/𝑝))^(1/𝑞)
Matrix L₁₁/Absolute Norm	NORM	matrices	𝐿₁₁ = \|\|𝐴\|\|₁₁ = (𝛴_𝑗(𝛴_𝑖\|𝐴_𝑖𝑗\|¹)^(1/1))^(1/1) = 𝛴_𝑖𝑗\|𝐴_𝑖𝑗\|
Matrix L22 L𝐹 Frobenius Norm	NORM	matrices	𝐿₂₂ = \|\|𝐴\|\|₂₂ = (𝛴_𝑗(𝛴_𝑖\|𝐴_𝑖𝑗\|²)^(2/2))^(1/2)= [𝛴_𝑖𝑗\|𝐴_𝑖𝑗\|²]^(1/2) = 𝐿_𝐹
Matrix L_∞∞/Max/Tchebychev Norm	NORM	matrices	𝐿_∞∞ = \|\|𝐴\|\|_∞∞ = (𝛴_𝑗(𝛴_𝑖\|𝐴_𝑖𝑗\|^∞)^(∞/∞))^(1/∞) = 𝑚𝑎𝑥(𝐴_𝑖𝑗)
Operator Norm	NORM	matrices	informally, the operator norm ‖𝐿‖_𝑜𝑝 of a linear operator 𝐿: 𝑋→𝑌 is the maximum factor by which it “lengthens” vectors
Spectral Norm	NORM	matrices	TODO

Type	Category	Input	Description
Supremum Norm - Sup Norm - Uniform Norm	NORM	depends	\|\|𝑓\|\|_∞ = \|\|𝑓\|\|_∞,𝑆 = 𝑠𝑢𝑝{\|𝑓(𝑠)\| : 𝑠∊𝑆}
Inner Product Norm	NORM	depends	an inner product (⟨·,·⟩) induces a norm (\|\|·\|\|_⟨·,·⟩) defined as: \|\|·\|\|_⟨·,·⟩ = √⟨·,·⟩
Constant 0 Map	SEMI-NORM	anything	the trivial semi-norm on 𝑉 which refers to the constant 0 map on 𝑉