Group Invariant Bounded Linear Functions on Dedekind Complete Totally Ordered Riesz Spaces

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Abstract

In this paper we consider the set B of all bounded subsets of V, where V is a totally ordered Dedekind complete Riesz space equipped with the order topology. We show the existence of bounded linear functions on B that are invariant under group actions of the symmetric group of B
.To do this, we construct a set of “approximately” group
invariant bounded linear functions and we show, using Tychonff’s Theorem (that is equivalent to the Axiom of Choice), that this set has a cluster point. This cluster point is the group invariant bounded linear function on B that we are looking for.
Original languageEnglish
Pages (from-to)206-211
Number of pages6
JournalAsian Journal of Mathematics and Computer Research
Volume7
Issue number4
Publication statusPublished - Apr 2017

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Riesz Space
Ordered Space
Linear Function
Invariant
Order Topology
Axiom of choice
Group Action
Symmetric group
Subset
Theorem

Keywords

  • Riesz spaces
  • Tychonoff's Theorem
  • Group Actions
  • Bounded Linear Functions

Cite this

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title = "Group Invariant Bounded Linear Functions on Dedekind Complete Totally Ordered Riesz Spaces",
abstract = "In this paper we consider the set B of all bounded subsets of V, where V is a totally ordered Dedekind complete Riesz space equipped with the order topology. We show the existence of bounded linear functions on B that are invariant under group actions of the symmetric group of B.To do this, we construct a set of “approximately” group invariant bounded linear functions and we show, using Tychonff’s Theorem (that is equivalent to the Axiom of Choice), that this set has a cluster point. This cluster point is the group invariant bounded linear function on B that we are looking for.",
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author = "George Chailos",
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N2 - In this paper we consider the set B of all bounded subsets of V, where V is a totally ordered Dedekind complete Riesz space equipped with the order topology. We show the existence of bounded linear functions on B that are invariant under group actions of the symmetric group of B.To do this, we construct a set of “approximately” group invariant bounded linear functions and we show, using Tychonff’s Theorem (that is equivalent to the Axiom of Choice), that this set has a cluster point. This cluster point is the group invariant bounded linear function on B that we are looking for.

AB - In this paper we consider the set B of all bounded subsets of V, where V is a totally ordered Dedekind complete Riesz space equipped with the order topology. We show the existence of bounded linear functions on B that are invariant under group actions of the symmetric group of B.To do this, we construct a set of “approximately” group invariant bounded linear functions and we show, using Tychonff’s Theorem (that is equivalent to the Axiom of Choice), that this set has a cluster point. This cluster point is the group invariant bounded linear function on B that we are looking for.

KW - Riesz spaces

KW - Tychonoff's Theorem

KW - Group Actions

KW - Bounded Linear Functions

M3 - Article

VL - 7

SP - 206

EP - 211

JO - Asian Journal of Mathematics and Computer Research

JF - Asian Journal of Mathematics and Computer Research

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ER -