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. doi:10.7907/HPNF-KH28. https://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 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 σ-Dedekind 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 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 T_n of the operator θ ≦ T ∈ L_b(L,M) is given by T_n(u) = inf {sup _αT(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 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 L_ρ is a normed Riesz space with the (sequential) Fatou property, then L^_ρ^ has the (sequential) Fatou property.

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