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 ${\displaystyle K>0.}$

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

References

• Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. Springer. ISBN 0-7923-6970-X.
• Conway, John B. (1990). A Course in Functional Analysis. Springer. ISBN 0-387-97245-5.
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. 19. Springer. ISBN 3-540-50584-9.
• Swartz, Charles (1992). An Introduction to Functional Analysis. CRC Press. ISBN 0-8247-8643-2.