Quasinorm

Not to be confused with seminorm or pseudonorm.

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 ${\displaystyle K>1.}$

Definition

A quasi-seminorm ^[1] on a vector space ${\displaystyle X}$ is a real-valued map ${\displaystyle p}$ on ${\displaystyle X}$ that satisfies the following conditions:

1. Non-negativity: ${\displaystyle p\geq 0;}$
2. Absolute homogeneity : ${\displaystyle p(sx)=|s|p(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s;}$
3. there exists a real ${\displaystyle k\geq 1}$ such that ${\displaystyle p(x+y)\leq k[p(x)+p(y)]}$ for all ${\displaystyle x,y\in X.}$
   • If ${\displaystyle k=1}$ 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:

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

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

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

Topology

If ${\displaystyle p}$ is a quasinorm on ${\displaystyle X}$ then ${\displaystyle p}$ induces a vector topology on ${\displaystyle X}$ whose neighborhood basis at the origin is given by the sets: ^[2]

$${\displaystyle \{x\in X:p(x)<1/n\}}$$

as ${\displaystyle n}$ 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

See also: Banach algebra

A quasinormed space ${\displaystyle (A,\|\,\cdot \,\|)}$ is called a quasinormed algebra if the vector space ${\displaystyle A}$ is an algebra and there is a constant ${\displaystyle K>0}$ 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. ^[2]

Examples

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

${\displaystyle L^{p}}$ spaces with ${\displaystyle 0<p<1}$

The ${\displaystyle L^{p}}$ spaces for ${\displaystyle 0<p<1}$ 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 ${\displaystyle 0<p<1,}$ the Lebesgue space ${\displaystyle L^{p}([0,1])}$ is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself ${\displaystyle L^{p}([0,1])}$ and the empty set) and the only continuous linear functional on ${\displaystyle L^{p}([0,1])}$ is the constant ${\displaystyle 0}$ function ( Rudin 1991 , §1.47). In particular, the Hahn-Banach theorem does not hold for ${\displaystyle L^{p}([0,1])}$ when ${\displaystyle 0<p<1.}$

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 homogenousPages displaying wikidata descriptions as a fallback
• Topological vector space  – Vector space with a notion of nearness

References

1. Kalton 1986, pp. 297–324.
2. Wilansky 2013, p. 55.

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• Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces" (PDF). Studia Mathematica. Institute of Mathematics, Polish Academy of Sciences. 84 (3): 297–324. doi:10.4064/sm-84-3-297-324. ISSN   0039-3223.
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. Vol. 19. Springer. ISBN   3-540-50584-9.
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• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN   978-0-486-49353-4. OCLC   849801114.