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
Definition
A quasi-seminorm[1] on a vector space is a real-valued map on that satisfies the following conditions:
- Non-negativity:
- Absolute homogeneity: for all and all scalars
- there exists a real such that for all
- If then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.
A quasinorm[1] is a quasi-seminorm that also satisfies:
- Positive definite/Point-separating: if satisfies then
A pair consisting of a vector space and an associated quasi-seminorm is called a quasi-seminormed vector space. If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space.
Multiplier
The
A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
Topology
If is a quasinorm on then induces a vector topology on whose neighborhood basis at the origin is given by the sets:[2]
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a quasi-Banach space. Every Banach space is a quasi-Banach space, although not conversely.
Related definitions
A quasinormed space is called a quasinormed algebra if the vector space is an algebra and there is a constant such that
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.[2]
Examples
Since every norm is a quasinorm, every
spaces with
The spaces for are quasinormed spaces (indeed, they are even
See also
- Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
- Norm (mathematics) – Length in a vector space
- Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous
- Topological vector space – Vector space with a notion of nearness
References
- ^ a b Kalton 1986, pp. 297–324.
- ^ a b Wilansky 2013, p. 55.
- Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. ISBN 0-7923-6970-X.
- Conway, John B. (1990). A Course in Functional Analysis. ISBN 0-387-97245-5.
- Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces" (PDF). Studia Mathematica. 84 (3). Institute of Mathematics, Polish Academy of Sciences: 297–324. ISSN 0039-3223.
- Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. Vol. 19. ISBN 3-540-50584-9.
- OCLC 21163277.
- Swartz, Charles (1992). An Introduction to Functional Analysis. ISBN 0-8247-8643-2.
- OCLC 849801114.