In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by
‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)}
for some K > 1. {\displaystyle K>1.}
Definition A quasi-seminorm on a vector space X {\displaystyle X} is a real-valued map p {\displaystyle p} on X {\displaystyle X} that satisfies the following conditions:
Non-negativity: p ≥ 0 ; {\displaystyle p\geq 0;}
Absolute homogeneity: p ( s x ) = | s | p ( x ) {\displaystyle p(sx)=|s|p(x)} for all x ∈ X {\displaystyle x\in X} and all scalars s ; {\displaystyle s;}
there exists a real k ≥ 1 {\displaystyle k\geq 1} such that p ( x + y ) ≤ k [ p ( x ) + p ( y ) ] {\displaystyle p(x+y)\leq k[p(x)+p(y)]} for all x , y ∈ X . {\displaystyle x,y\in X.}
If k = 1 {\displaystyle k=1} then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.
A quasinorm is a quasi-seminorm that also satisfies:
Positive definite/Point-separating: if x ∈ X {\displaystyle x\in X} satisfies p ( x ) = 0 , {\displaystyle p(x)=0,} then x = 0. {\displaystyle x=0.}
A pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and an associated quasi-seminorm p {\displaystyle p} is called a quasi-seminormed vector space. If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space. Multiplier The infimum of all values of k {\displaystyle k} that satisfy condition (3) is called the multiplier of p . {\displaystyle p.} The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term k {\displaystyle k} -quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to k . {\displaystyle k.} A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1. {\displaystyle 1.} Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
Topology If p {\displaystyle p} is a quasinorm on X {\displaystyle X} then p {\displaystyle p} induces a vector topology on X {\displaystyle X} whose neighborhood basis at the origin is given by the sets:
{ x ∈ X : p ( x ) < 1 / n } {\displaystyle \{x\in X:p(x)<1/n\}}
as n {\displaystyle n} ranges over the positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space. Every quasinormed topological vector space is pseudometrizable. A complete quasinormed space is called a quasi-Banach space. Every Banach space is a quasi-Banach space, although not conversely.
Related definitions
A quasinormed space ( A , ‖ ⋅ ‖ ) {\displaystyle (A,\|\,\cdot \,\|)} is called a quasinormed algebra if the vector space A {\displaystyle A} is an algebra and there is a constant K > 0 {\displaystyle K>0} such that
‖ x y ‖ ≤ K ‖ x ‖ ⋅ ‖ y ‖ {\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}
for all x , y ∈ A . {\displaystyle x,y\in A.}
A complete quasinormed algebra is called a quasi-Banach algebra.
Characterizations A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.
Examples Since every norm is a quasinorm, every normed space is also a quasinormed space.
L p {\displaystyle L^{p}} spaces with 0 < p < 1 {\displaystyle 0<p<1}
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