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 $K > 1.$

This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satisfied except for positive definiteness.

Related concepts

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

A complete quasinormed vector 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.

See also

• Seminorm