In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and related areas of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quasinorm is similar to a norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 in that it satisfies the norm axioms, except that the triangle inequality

Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....

 is replaced by
for some

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

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 with an associated quasinorm is called a quasinormed vector space.

A complete

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 quasinormed vector space is called a quasi-Banach space.

A quasinormed space is called a quasinormed algebra if the vector space A is an algebra

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

and there is a constant K > 0 such that
for all .

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