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 Template:EmTemplate:Sfn on a vector space $X$ is a real-valued map $p$ on $X$ that satisfies the following conditions:

1. Template:Em: $p\ge 0;$
2. Template:Em: $p(sx)=|s|p(x)$ 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 Template:EmTemplate:Sfn is a quasi-seminorm that also satisfies:

4. Positive definite/Template:Visible anchor: 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 Template:Em. If the quasi-seminorm is a quasinorm then it is also called a Template:Em.

Multiplier

The infimum of all values of $k$ that satisfy condition (3) is called the Template:Em 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 Template:Em is sometimes used to describe a quasi-seminorm whose multiplier is equal to $k.$

A Template:Em (respectively, a Template:Em) 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:Template:Sfn $$\{x\in X:p(x)<1/n\}$$ as $n$ ranges over the positive integers. A topological vector space with such a topology is called a Template:Em or just a Template:Em.

Every quasinormed topological vector space is pseudometrizable.

A complete quasinormed space is called a Template:Em. Every Banach space is a quasi-Banach space, although not conversely.

Related definitions

A quasinormed space $(A,\Vert \cdot \Vert )$ is called a Template:Em 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 Template:Em.

Characterizations

A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.Template:Sfn

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 Template:Em locally convex (in fact, its only convex open subsets are itself $L^p([0,1])$ and the empty set) and the Template:Em continuous linear functional on $L^p([0,1])$ is the constant $0$ function Template:Harv. In particular, the Hahn-Banach theorem does Template:Em hold for $L^p([0,1])$ when $0<p<1.$

See also

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References

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• Template:Rudin Walter Functional Analysis
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• Template:Wilansky Modern Methods in Topological Vector Spaces