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:09192018073917103
Abstract
Chapter 1 contains a summary of results on Riesz spaces frequently used in this thesis.
Chapter 2 considers the real linear space L_{b}(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 L_{b}(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 L^{e} _{b}(A, M). Two theorems are proved:
(i) If θ ≦ T is extendable, then T has a smallest positive extension T_{m}_{'} given by T_{m}(u) = sup {T(v): v ∈ A; θ ≦ v ≦ u} for all u in L^{+}.
(ii) The mapping T →(T^{+})_{m}  (T^{})_{m} from L^{e}_{b}(A, M) into L_{b}(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 ∈ L^{e} _{b}(A,M) is a normal integral, then so is T_{m}.
(ii) If L is σDedekind complete and M is super Dedekind complete, then T in L_{b}(L,M) is a normal integral if and only if N_{T} = {u ∈ L: T (u) = θ} is a band of L.
(iii) If L is σDedekind 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 T_{n} of the operator θ ≦ T ∈ L_{b}(L,M) is given by T_{n}(u) = inf {sup _{α}T(u_{α}): θ ≦ u_{α} ↑ 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 τclosed ideal is a σideal, and (L, τ) satisfies (A,ii) if every τclosed 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 nonmetrizable 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 L_{ρ} 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)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  18 April 1973  
Funders: 
 
Record Number:  CaltechTHESIS:09192018073917103  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:09192018073917103  
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  
Last Modified:  21 Sep 2018 18:45 
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