REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES

by Neil E. Gretsky

University of California, Riverside

CHAPTER I: INTRODUCTION

1. The Problem. The Banach function space L (0,S,|i) is

a Banach space of (equivalence classes of) measurable functions

on the measure space (Q,£,|-t) with p as a (function) norm.

Banach function spaces, sometimes called Riesz spaces or Kothe-

Toeplitz spaces have been studied in [16, 34, 14, 15, 7, 8, 18,

19, 21, 22, 23] with the most exhaustive work being in [20]. They

include many of the well-known function spaces such as the Lebesgue

spaces and the Orlicz spaces and are well suited for settings

in analysis for work in many areas, e.g. integral equations and

stochastic transformations.

The problems considered in this paper are the obtaining of

integral representations of members of B(£,L ) and of B(L ,£)

5

where X is an arbitrary Banach space and (B(ty,Z) denotes the

spaces of continuous linear operators from I j to Z . In Chapter

II, under the hypothesis that the function norm satisfies a mild

type of averaging condition, a characterization of members of the

space 8(£,L ) is obtained as an integral representation in terms

of £ -valued additive set functions of p-bounded variation. The

result here generalizes the fundamental representation theorems of

Dunford and Pettis [5] and their later generalizations (cf [6] ,

p. 498). In the second section of Chapter II, using the same

This paper constitutes the major portion of the author's

doctoral dissertation written at Carnegie Institute of Technology

under the guidance of Professor M. M. Rao. The author acknowledges

a deep debt of gratitude to Dr. Rao.

Received by the editor 10-25-67 and,in revised form,7-19-68.