John Annulis

Publication Date

7-26-1971

Abstract

We study decompositions of the identity (for definition see Vulikh [13]) in Dedekind a-complete vector lattices. In Chapter 1, we define and develop a notion of continuous elements with respect to a decomposition of the identity. We shall also develop a spectral theory for these elements. In Chapter 2, two Stieltjes-type type integrals are defined with respect to decompositions of the identity. The relationship of these integrals to the continuous elements is then explored. In Chapter 3, we prove two results in ordered vector spaces. The first result is that every infinite dimensional Dedekind complete vector lattice has a base with an infinite number of elements. The second result is a characterization of extended spaces in terms of decompositions of the identity. Consequences of these two theorems are then explored. Chapter 4 is devoted to applications and examples of the preceding three chapters. Included in Chapter 4 is a version of the Riesz Representation Theorem for ordered vector spaces.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Richard Clyde Metzler

Second Committee Member

Donald Ward Dubois

Third Committee Member

Illegible

Language

English

Document Type

Dissertation

Download

Share

COinS