View source for Quasinorm

{{confused|seminorm|pseudonorm}}

In [[linear algebra]], [[functional analysis]] and related areas of [[mathematics]], a '''quasinorm''' is similar to a [[Norm (mathematics)|norm]] in that it satisfies the norm axioms, except that the [[triangle inequality]] is replaced by
<math display=block>\|x + y\| \leq K(\|x\| + \|y\|)</math>
for some <math>K > 1.</math>

==Definition==

A {{em|{{visible anchor|quasi-seminorm|quasi seminorm|quasi semi-norm|text=quasi-seminorm}}}}{{sfn|Kalton|1986|pp=297–324}} on a vector space <math>X</math> is a real-valued map <math>p</math> on <math>X</math> that satisfies the following conditions:
<ol>
<li>{{em|Non-negativity}}: <math>p \geq 0;</math></li>
<li>{{em|[[Homogeneous function|Absolute homogeneity]]}}: <math>p(s x) = |s| p(x)</math> for all <math>x \in X</math> and all scalars <math>s;</math></li>
<li>there exists a real <math>k \geq 1</math> such that <math>p(x + y) \leq k [p(x) + p(y)]</math> for all <math>x, y \in X.</math>
* If <math>k = 1</math> then this inequality reduces to the [[triangle inequality]]. It is in this sense that this condition generalizes the usual triangle inequality.</li>
</ol>

A {{em|{{visible anchor|quasi-norm|quasi norm|quasinorm|text=quasinorm}}}}{{sfn|Kalton|1986|pp=297–324}} is a quasi-seminorm that also satisfies:
<ol start=4>
<li>[[Positive definite]]/{{visible anchor|Point-separating}}: if <math>x \in X</math> satisfies <math>p(x) = 0,</math> then <math>x = 0.</math></li>
</ol>

A pair <math>(X, p)</math> consisting of a [[vector space]] <math>X</math> and an associated quasi-seminorm <math>p</math> is called a {{em|{{visible anchor|quasi-seminormed vector space|quasi seminormed vector space|quasi semi-normed vector space|text=quasi-seminormed vector space}}}}. 
If the quasi-seminorm is a quasinorm then it is also called a {{em|{{visible anchor|quasi-normed vector space|quasi normed vector space|quasinormed vector space|text=quasinormed vector space}}}}. 

'''Multiplier'''

The [[infimum]] of all values of <math>k</math> that satisfy condition (3) is called the {{em|{{visible anchor|multiplier}}}} of <math>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 {{em|<math>k</math>-quasi-seminorm}} is sometimes used to describe a quasi-seminorm whose multiplier is equal to <math>k.</math> 

A {{em|[[Norm (mathematics)|norm]]}} (respectively, a {{em|[[seminorm]]}}) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is <math>1.</math> 
Thus every [[seminorm]] is a quasi-seminorm and every [[Norm (mathematics)|norm]] is a quasinorm (and a quasi-seminorm).

===Topology===

If <math>p</math> is a quasinorm on <math>X</math> then <math>p</math> induces a vector topology on <math>X</math> whose neighborhood basis at the origin is given by the sets:{{sfn|Wilansky|2013|p=55}}
<math display=block>\{x \in X : p(x) < 1/n\}</math>
as <math>n</math> ranges over the positive integers. 
A [[topological vector space]] with such a topology is called a {{em|{{visible anchor|quasinormed topological vector space}}}} or just a {{em|{{visible anchor|quasinormed space}}}}.

Every quasinormed topological vector space is [[Metrizable topological vector space|pseudometrizable]]. 

A [[Complete topological vector space|complete]] quasinormed space is called a {{em|{{visible anchor|quasi-Banach space}}}}. Every [[Banach space]] is a quasi-Banach space, although not conversely. 

===Related definitions===

{{See also|Banach algebra}}

A quasinormed space <math>(A, \| \,\cdot\, \|)</math> is called a {{em|{{visible anchor|quasinormed algebra}}}} if the vector space <math>A</math> is an [[Algebra over a field|algebra]] and there is a constant <math>K > 0</math> such that 
<math display=block>\|x y\| \leq K \|x\| \cdot \|y\|</math>
for all <math>x, y \in A.</math>

A [[Complete topological vector space|complete]] quasinormed algebra is called a {{em|{{visible anchor|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.{{sfn|Wilansky|2013|p=55}}

==Examples==

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

'''<math>L^p</math> spaces with <math>0 < p < 1</math>'''

The [[Lp space|<math>L^p</math> spaces]] for <math>0 < p < 1</math> are quasinormed spaces (indeed, they are even [[F-space]]s) but they are not, in general, [[Normable space|normable]] (meaning that there might not exist any norm that defines their topology). 
For <math>0 < p < 1,</math> the [[Lp space|Lebesgue space]] <math>L^p([0, 1])</math> is a [[Complete metric space|complete]] [[Metrizable topological vector space|metrizable TVS]] (an [[F-space]]) that is {{em|not}} [[Locally convex topological vector space|locally convex]] (in fact, its only [[Convex set|convex]] open subsets are itself <math>L^p([0, 1])</math> and the empty set) and the {{em|only}} [[continuous linear functional]] on <math>L^p([0, 1])</math> is the constant <math>0</math> function {{harv|Rudin|1991|loc=§1.47}}. 
In particular, the [[Hahn-Banach theorem]] does {{em|not}} hold for <math>L^p([0, 1])</math> when <math>0 < p < 1.</math> 

==See also==

* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Norm (mathematics)}}
* {{annotated link|Seminorm}}
* {{annotated link|Topological vector space}}

==References==

{{reflist}}
{{reflist|group=note}}

* {{cite book|last=Aull|first=Charles E.|author2=Robert Lowen|title=Handbook of the History of General Topology|year=2001|isbn=0-7923-6970-X|publisher=[[Springer Science+Business Media|Springer]]}}
* {{cite book|last=Conway|first=John B.|title=A Course in Functional Analysis|isbn=0-387-97245-5|publisher=[[Springer Science+Business Media|Springer]]|year=1990}}
* {{cite journal|last=Kalton|first=N.|title=Plurisubharmonic functions on quasi-Banach spaces|journal=Studia Mathematica|publisher=Institute of Mathematics, Polish Academy of Sciences|volume=84|issue=3|year=1986|issn=0039-3223|doi=10.4064/sm-84-3-297-324|pages=297–324|url=https://kaltonmemorial.missouri.edu/assets/docs/sm1986b.pdf}} <!--{{sfn|Kalton|1986|pp=297–324}}-->
* {{cite book|last=Nikolʹskiĭ|first=Nikolaĭ Kapitonovich|title=Functional Analysis I: Linear Functional Analysis|isbn=3-540-50584-9|year=1992|publisher=[[Springer Science+Business Media|Springer]]|series=Encyclopaedia of Mathematical Sciences|volume=19}}
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{cite book|last=Swartz|first=Charles|title=An Introduction to Functional Analysis|year=1992|publisher=[[CRC Press]]|isbn=0-8247-8643-2}}
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!--{{sfn|Wilansky|2013|p=}}-->

{{Lp spaces}}
{{Functional analysis}}
{{Topological vector spaces}}
{{Banach spaces}}

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[[Category:Linear algebra]]
[[Category:Norms (mathematics)]]