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

${\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)}$

for some K > 0.

Related concepts

Definition:[1] 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 x ∈ X and all scalars s;
3. there exists a k ≥ 1 such that p(x + y) ≤ k[p(x) + p(y)] for all x, y ∈ X.

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:[1]

{ x ∈ X : 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 ${\displaystyle (A,\|\cdot \|)}$ is called a quasinormed algebra if the vector space A is an algebra and there is a constant K > 0 such that

${\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}$

for all ${\displaystyle x,y\in A}$.

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.[1]

See also

• Metrizable TVS
• Seminorm
• Topological vector space – Vector space with a notion of nearness

References

1. Wilansky 2013, p. 55.

• 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.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.