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# On Order and Topological Properties of Riesz Spaces

## Citation

Aliprantis, Charalambos Dionisios (1973) On Order and Topological Properties of Riesz Spaces. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:09192018-073917103

## Abstract

Chapter 1 contains a summary of results on Riesz spaces frequently used in this thesis.

Chapter 2 considers the real linear space Lb(L, M) of all order bounded linear transformations from a Riesz space L into a Dedekind complete Riesz space M. The order structure of the Dedekind complete Riesz space Lb(L, M) is studied in some detail. Dual formulas for T(f+), T(f-) and T(|f|) are proved. The linear space of all extendable operators from the ideal A of L into M is denoted by Le b(A, M). Two theorems are proved:

(i) If θ ≦ T is extendable, then T has a smallest positive extension Tm' given by Tm(u) = sup {T(v): v ∈ A; θ ≦ v ≦ u} for all u in L+.

(ii) The mapping T →(T+)m - (T-)m from Leb(A, M) into Lb(L, M) is a Riesz isomorphism.

Chapter 3 studies integral and normal integral transformations. Some of the theorems included in this chapter are:

(i) If T ∈ Le b(A,M) is a normal integral, then so is Tm.

(ii) If L is σ-Dedekind complete and M is super Dedekind complete, then T in Lb(L,M) is a normal integral if and only if NT = {u ∈ L: |T |(|u|) = θ} is a band of L.

(iii) If L is σ-Dedekind complete and M is super Dedekind complete and if there exists a strictly positive operator for L into M, then L is super Dedekind complete.

(iv) If M admits a strictly positive linear functional which is normal then the normal component Tn of the operator θ ≦ T ∈ Lb(L,M) is given by Tn(u) = inf {sup αT(uα): θ ≦ uα ↑ u} for all u in L+.

Chapter 4 studies ordered topological vector spaces (E,τ) with particular emphasis on locally solid linear topological Riesz spaces. Order continuity and topological continuity are considered by introducing the properties (A,o), (A,i), (A,ii), (A,iii) and (A,iv). Some results from this chapter are:

(i) If (L, τ) is a locally solid Riesz space, then (L,τ) satisfies (A,i) if every τ-closed ideal is a σ-ideal, and (L, τ) satisfies (A,ii) if every τ-closed ideal is a band.

(ii) If (L,τ) is a metrizable locally solid Riesz space with (A,ii), then L satisfies the Egoroff property.

(iii) If (L,τ) is a metrizable locally solid Riesz space, then both (A,i) and (A,iii) hold if (A,ii) holds. A counter example shows that this is not true for non-metrizable locally solid Riesz spaces.

The fifth and final chapter considers Hausdorff locally solid Riesz spaces (L, τ). The topological completion of (L, τ) is denoted by (L^, τ^). Some results from this chapter are:

(i) (L^,τ^) is a Hausdorff locally solid Riesz space with cone L^+ = L+ = the τ^-closure of L + in L^, containing L as a Riesz subspace.

(ii) (L^,τ^) satisfies the (A,iii) property, if (L, τ) does.

(iii) (L^,τ^) satisfies the (A,ii) property, if (L, τ) does.

(iv) If τ is metrizable, then (L^,τ^) satisfies the (A,i) property if (L, τ) does.

(v) If Lρ is a normed Riesz space with the (sequential) Fatou property, then L^ρ^ has the (sequential) Fatou property.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
• Luxemburg, W. A. J.
Thesis Committee:
• Unknown, Unknown
Defense Date:18 April 1973
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:09192018-073917103
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:09192018-073917103
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11189
Collection:CaltechTHESIS
Deposited By: Tony Diaz
Deposited On:21 Sep 2018 18:45  Preview