# Quasinorm

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 $\|x+y\|\leq K(\|x\|+\|y\|)$ for some $K>1.$

## Definition

A quasi-seminorm on a vector space $X$ is a real-valued map $p$ on $X$ that satisfies the following conditions:
* Non-negativity: $p\geq 0;$
* Absolute homogeneity: $p(sx)=|s|p(x)$ for all $x\in X$ and all scalars $s;$
* there exists a real $k\geq 1$ such that $p(x+y)\leq k[p(x)+p(y)]$ for all $x,y\in X.$
** If $k=1$ 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 $x\in X$ satisfies $p(x)=0,$ then $x=0.$
A pair $(X,p)$ consisting of a vector space $X$ and an associated quasi-seminorm $p$ 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 $k$ that satisfy condition (3) is called the multiplier of $p.$ The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term $k$-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to $k.$
A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is $1.$ Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).

### Topology

If $p$ is a quasinorm on $X$ then $p$ induces a vector topology on $X$ whose neighborhood basis at the origin is given by the sets: $\{x\in X:p(x)<1/n\}$ as $n$ 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 $(A,\|\,\cdot \,\|)$ is called a quasinormed algebra if the vector space $A$ is an algebra and there is a constant $K>0$ such that $\|xy\|\leq K\|x\|\cdot \|y\|$ for all $x,y\in A.$
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.
$L^{p}$ spaces with $0<p<1$
The $L^{p}$ spaces for $0<p<1$ are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For $0<p<1,$ the Lebesgue space $L^{p}([0,1])$ is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself $L^{p}([0,1])$ and the empty set) and the only continuous linear functional on $L^{p}([0,1])$ is the constant $0$ function (Rudin 1991, §1.47). In particular, the Hahn-Banach theorem does not hold for $L^{p}([0,1])$ when $0<p<1.$