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 [math]\displaystyle{ \|x + y\| \leq K(\|x\| + \|y\|) }[/math] for some [math]\displaystyle{ K \gt 1. }[/math]

Definition

A quasi-seminorm on a vector space [math]\displaystyle{ X }[/math] is a real-valued map [math]\displaystyle{ p }[/math] on [math]\displaystyle{ X }[/math] that satisfies the following conditions:

1. Non-negativity: [math]\displaystyle{ p \geq 0; }[/math]
2. Absolute homogeneity: [math]\displaystyle{ p(s x) = |s| p(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all scalars [math]\displaystyle{ s; }[/math]
3. there exists a real [math]\displaystyle{ k \geq 1 }[/math] such that [math]\displaystyle{ p(x + y) \leq k [p(x) + p(y)] }[/math] for all [math]\displaystyle{ x, y \in X. }[/math]
   • If [math]\displaystyle{ k = 1 }[/math] then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.

A quasinorm is a quasi-seminorm that also satisfies:

4. Positive definite/Point-separating: if [math]\displaystyle{ x \in X }[/math] satisfies [math]\displaystyle{ p(x) = 0, }[/math] then [math]\displaystyle{ x = 0. }[/math]

A pair [math]\displaystyle{ (X, p) }[/math] consisting of a vector space [math]\displaystyle{ X }[/math] and an associated quasi-seminorm [math]\displaystyle{ p }[/math] 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 infimum of all values of [math]\displaystyle{ k }[/math] that satisfy condition (3) is called the multiplier of [math]\displaystyle{ p. }[/math] The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term [math]\displaystyle{ k }[/math]-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to [math]\displaystyle{ k. }[/math]

A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is [math]\displaystyle{ 1. }[/math] Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).

Topology

If [math]\displaystyle{ p }[/math] is a quasinorm on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ p }[/math] induces a vector topology on [math]\displaystyle{ X }[/math] whose neighborhood basis at the origin is given by the sets: [math]\displaystyle{ \{x \in X : p(x) \lt 1/n\} }[/math] as [math]\displaystyle{ n }[/math] ranges over the positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space.

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 [math]\displaystyle{ (A, \| \,\cdot\, \|) }[/math] is called a quasinormed algebra if the vector space [math]\displaystyle{ A }[/math] is an algebra and there is a constant [math]\displaystyle{ K \gt 0 }[/math] such that [math]\displaystyle{ \|x y\| \leq K \|x\| \cdot \|y\| }[/math] for all [math]\displaystyle{ x, y \in A. }[/math]

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.

Examples

Since every norm is a quasinorm, every normed space is also a quasinormed space.

[math]\displaystyle{ L^p }[/math] spaces with [math]\displaystyle{ 0 \lt p \lt 1 }[/math]

The [math]\displaystyle{ L^p }[/math] spaces for [math]\displaystyle{ 0 \lt p \lt 1 }[/math] are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For [math]\displaystyle{ 0 \lt p \lt 1, }[/math] the Lebesgue space [math]\displaystyle{ L^p([0, 1]) }[/math] is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself [math]\displaystyle{ L^p([0, 1]) }[/math] and the empty set) and the only continuous linear functional on [math]\displaystyle{ L^p([0, 1]) }[/math] is the constant [math]\displaystyle{ 0 }[/math] function (Rudin 1991). In particular, the Hahn-Banach theorem does not hold for [math]\displaystyle{ L^p([0, 1]) }[/math] when [math]\displaystyle{ 0 \lt p \lt 1. }[/math]

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
• Topological vector space – Vector space with a notion of nearness

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. 
• Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces". Studia Mathematica (Institute of Mathematics, Polish Academy of Sciences) 84 (3): 297–324. doi:10.4064/sm-84-3-297-324. ISSN 0039-3223. https://kaltonmemorial.missouri.edu/assets/docs/sm1986b.pdf. 
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. 19. Springer. ISBN 3-540-50584-9. 
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• 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.