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.
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