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| 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
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| :<math>\|x + y\| \leq K(\|x\| + \|y\|)</math> | |
| for some <math>K > 1.</math>
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| This is not to be confused with a '''[[seminorm]]''' or '''[[pseudonorm]]''', where the norm axioms are satisfied except for positive definiteness.
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| == Related concepts == | |
| A [[vector space]] with an associated quasinorm is called a '''quasinormed vector space'''.
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| A [[complete space|complete]] quasinormed vector space is called a '''quasi-Banach space'''.
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| A quasinormed space <math>(A, \| \cdot \|)</math> is called a '''quasinormed algebra''' if the vector space ''A'' is an [[Algebra over a field|algebra]] and there is a constant ''K'' > 0 such that
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| :<math>\|xy\| \leq K \|x\| \cdot \|y\|</math> | |
| for all <math>x, y \in A</math>.
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| A complete quasinormed algebra is called a '''quasi-Banach algebra'''.
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| == See also ==
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| *[[Seminorm]]
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| == References ==
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| * {{cite book | title=Handbook of the History of General Topology | last=Aull | first=Charles E. | coauthors=Robert Lowen | year=2001 | isbn=0-7923-6970-X | publisher=[[Springer Science+Business Media|Springer]] }}
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| * {{cite book | title=A Course in Functional Analysis | last=Conway | first=John B. | isbn=0-387-97245-5 | publisher=[[Springer Science+Business Media|Springer]] | year=1990 }}
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| * {{cite book | title=Functional Analysis I: Linear Functional Analysis | last=Nikolʹskiĭ | first=Nikolaĭ Kapitonovich | isbn=3-540-50584-9 | year=1992 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences | volume=19 }}
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| * {{cite book | title=An Introduction to Functional Analysis | last=Swartz | first=Charles | year=1992 | publisher=[[CRC Press]] | isbn=0-8247-8643-2 }}
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| <!--Categories-->
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| [[Category:Norms (mathematics)]]
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| [[Category:Linear algebra]]
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