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In [[functional analysis]] and related areas of [[mathematics]] an '''absorbing set''' in a [[vector space]] is a [[Set (mathematics)|set]] ''S'' which can be ''inflated'' to include any element of the vector space. Alternative terms are '''[[radial set|radial]]''' or '''absorbent set'''. | |||
==Definition== | |||
Given a vector space ''X'' over the [[field (mathematics)|field]] '''F''' of [[Real number|real]] or [[Complex number|complex]] numbers, a set ''S'' is called '''absorbing''' if for all <math>x\in X</math> there exists a real number ''r'' such that | |||
:<math>\forall \alpha \in \mathbb{F} : \vert \alpha \vert \ge r \Rightarrow x \in \alpha S</math> | |||
with | |||
:<math>\alpha S := \{ \alpha s \mid s \in S\}</math> | |||
== Examples == | |||
*In a [[semi normed vector space]] the [[unit ball]] is absorbing. | |||
== Properties == | |||
*The finite [[intersection (set theory)|intersection]] of absorbing sets is absorbing | |||
==See also == | |||
*[[Algebraic interior]] | |||
*[[Bounded set (topological vector space)]] | |||
==References== | |||
* {{cite book |last=Robertson |first=A.P. |coauthors= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | page=4}} | |||
* {{cite book | last = Schaefer | first = Helmuth H. | year = 1971 | title = Topological vector spaces | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6 | page=11 }} | |||
{{Functional Analysis}} | |||
{{Mathanalysis-stub}} | |||
[[Category:Functional analysis]] | |||
[[fr:Ensemble absorbant]] | |||
Revision as of 02:27, 9 January 2014
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. Alternative terms are radial or absorbent set.
Definition
Given a vector space X over the field F of real or complex numbers, a set S is called absorbing if for all there exists a real number r such that
with
Examples
- In a semi normed vector space the unit ball is absorbing.
Properties
- The finite intersection of absorbing sets is absorbing
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