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In [[linear algebra]] and related areas of [[mathematics]] a '''balanced set''', '''circled set''' or '''disk''' in a [[vector space]] (over a [[field (mathematics)|field]] ''K'' with an [[absolute value (algebra)|absolute value]] |.|) is a [[Set (mathematics)|set]] ''S'' so that for all [[scalar (mathematics)|scalar]]s α with |α| ≤ 1 | |||
:<math>\alpha S \subseteq S</math> | |||
where | |||
:<math>\alpha S := \{\alpha x \mid x \in S\} .</math> | |||
The '''balanced hull''' or '''balanced envelope''' for a set ''S'' is the smallest balanced set containing ''S''. It can be constructed as the [[intersection (set theory)|intersection]] of all balanced sets containing ''S''. | |||
== Examples == | |||
* The open and closed [[open ball|balls]] in a [[normed vector space]] are balanced sets. | |||
* Any subspace of a real or complex vector space is a balanced set. | |||
* The [[cartesian product]] of a family of balanced sets is balanced in the [[product space]] of the corresponding vector spaces (over the same field ''K''). | |||
* Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do. As a result, ℂ and ℝ<sup>2</sup> are entirely different as far as their vector space structure is concerned. | |||
* If p is a semi-norm on a linear space X, then for any constant c>0, the set {x ∈ X | p(x)≤c} is balanced. | |||
== Properties == | |||
* The [[union (set theory)|union]] and [[intersection (set theory)|intersection]] of balanced sets is a balanced set. | |||
* The closure of a balanced set is balanced. | |||
* By definition (not property), a set is [[absolutely convex]] if and only if it is [[Convex set|convex]] and balanced. | |||
* Every balanced set is a [[symmetric set]] | |||
==See also== | |||
* [[Star domain]] | |||
==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 | author=W. Rudin | authorlink=Walter Rudin | title=Functional Analysis | edition=2nd ed | publisher=McGraw-Hill, Inc | date=1990 | isbn=0-07-054236-8 }} | |||
* {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | date=1970 | isbn=0-387-05380-8 | page=11 }} | |||
{{Functional Analysis}} | |||
[[Category:Linear algebra]] | |||
Revision as of 06:29, 14 March 2013
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K with an absolute value |.|) is a set S so that for all scalars α with |α| ≤ 1
where
The balanced hull or balanced envelope for a set S is the smallest balanced set containing S. It can be constructed as the intersection of all balanced sets containing S.
Examples
- The open and closed balls in a normed vector space are balanced sets.
- Any subspace of a real or complex vector space is a balanced set.
- The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field K).
- Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do. As a result, ℂ and ℝ2 are entirely different as far as their vector space structure is concerned.
- If p is a semi-norm on a linear space X, then for any constant c>0, the set {x ∈ X | p(x)≤c} is balanced.
Properties
- The union and intersection of balanced sets is a balanced set.
- The closure of a balanced set is balanced.
- By definition (not property), a set is absolutely convex if and only if it is convex and balanced.
- Every balanced set is a symmetric set
See also
References
- 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