# 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 > 0.

## Related concepts

Definition: A quasinorm on a vector space X is a real-valued map p on X that satisfies the following conditions:
1. Non-negativity: p ≥ 0;
2. Absolute homogeneity: p(sx) = |s| p(x) for all xX and all scalars s;
3. there exists a k ≥ 1 such that p(x + y) ≤ k[p(x) + p(y)] for all x, yX.

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:

{ xX : p(x) < 1/n}

as n ranges over the positive integers. A topological vector space (TVS) with such a topology is called a quasinormed space.

Every quasinormed TVS is a pseudometrizable.

A vector space with an associated quasinorm is called a quasinormed vector space.

A complete quasinormed space is called a quasi-Banach space.

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.