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 $$\Vert x+y\Vert \le K(\Vert x\Vert +\Vert y\Vert )$$ for some $K>1.$

Definition

A quasi-seminorm on a vector space $X$ is a real-valued map $p$ on $X$ that satisfies the following conditions:

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

A quasinorm is a quasi-seminorm that also satisfies:

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

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

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

Topology

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

A quasinormed space $(A,\Vert \cdot \Vert )$ is called a quasinormed algebra if the vector space $A$ is an algebra and there is a constant $K>0$ such that $$\Vert xy\Vert \le K\Vert x\Vert \cdot \Vert y\Vert$$ for all $x,y\in A.$

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.

Examples

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

$L^p$ spaces with $0<p<1$

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

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
• 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.