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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
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:<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]]

Latest revision as of 10:58, 12 January 2015


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