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https://hdl.handle.net/2142/71254
Description
Title
Integration and Differentiation in a Banach Space
Author(s)
Gordon, Russell Arthur
Issue Date
1987
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
The main focus of the original work in this paper is the extension of Saks's Theory of the Integral to functions that have values in a Banach space. The differentiation of functions that are not of bounded variation and the extension of the Denjoy integral to vector-valued functions are studied in detail. It is shown that a BVG$\sb\*$ function that has a measurable scalar derivative is differentiable almost everywhere, that the notions of weak differentiability almost everywhere and differentiability almost everywhere are equivalent, and that a BVG$\sb\*$ function that has values in a space with the Radon-Nikodym property is differentiable almost everywhere. Necessary and sufficient conditions for the existence of the Denjoy-Dunford integral are determined. It is shown that a space is weakly sequentially complete if and only if every measurable, Denjoy-Dunford integrable function is Denjoy-Pettis integrable. If X contains no copy of c$\sb{\rm O}$ and if f: (a,b) $\to$ X is Denjoy-Pettis integrable on (a,b), then every perfect set in (a,b) contains a portion on which f is Pettis integrable.
The Riemann integral of functions with values in a Banach space is discussed in detail in an expository chapter. The results of several authors are summarized. The classification of those Banach spaces for which Riemann integrability implies continuity almost everywhere is the highlight of this chapter. Two chapters deal with real-valued functions only. One presents the Denjoy integral while the other discusses the generalized Riemann integral. These chapters provide a good introduction to these integrals. A direct proof that the restricted Denjoy integral is equivalent to the generalized Riemann integral is given. Finally, a brief look at the generalized Riemann integral of vector-valued functions is included. For measurable functions this integral includes both the Pettis integral and the restricted Denjoy-Bochner integral.
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