Locally finite measure: Difference between revisions
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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 | |||
:<math>\|x + y\| \leq K(\|x\| + \|y\|)</math> | |||
for some <math>K > 1.</math> | |||
This is not to be confused with a '''[[seminorm]]''' or '''[[pseudonorm]]''', where the norm axioms are satisfied except for positive definiteness. | |||
== Related concepts == | |||
A [[vector space]] with an associated quasinorm is called a '''quasinormed vector space'''. | |||
A [[complete space|complete]] quasinormed vector space is called a '''quasi-Banach space'''. | |||
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 | |||
:<math>\|xy\| \leq K \|x\| \cdot \|y\|</math> | |||
for all <math>x, y \in A</math>. | |||
A complete quasinormed algebra is called a '''quasi-Banach algebra'''. | |||
== See also == | |||
*[[Seminorm]] | |||
== References == | |||
* {{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]] }} | |||
* {{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 }} | |||
* {{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 }} | |||
* {{cite book | title=An Introduction to Functional Analysis | last=Swartz | first=Charles | year=1992 | publisher=[[CRC Press]] | isbn=0-8247-8643-2 }} | |||
<!--Categories--> | |||
[[Category:Norms (mathematics)]] | |||
[[Category:Linear algebra]] | |||
Revision as of 16:48, 6 August 2013
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
for some
This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satisfied except for positive definiteness.
Related concepts
A vector space with an associated quasinorm is called a quasinormed vector space.
A complete quasinormed vector space is called a quasi-Banach space.
A quasinormed space is called a quasinormed algebra if the vector space A is an algebra and there is a constant K > 0 such that
for all .
A complete quasinormed algebra is called a quasi-Banach algebra.
See also
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534