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Title:  Integration and Differentiation in a Banach Space 
Author(s):  Gordon, Russell Arthur 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 vectorvalued 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 RadonNikodym property is differentiable almost everywhere. Necessary and sufficient conditions for the existence of the DenjoyDunford integral are determined. It is shown that a space is weakly sequentially complete if and only if every measurable, DenjoyDunford integrable function is DenjoyPettis integrable. If X contains no copy of c$\sb{\rm O}$ and if f: (a,b) $\to$ X is DenjoyPettis 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 realvalued 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 vectorvalued functions is included. For measurable functions this integral includes both the Pettis integral and the restricted DenjoyBochner integral. 
Issue Date:  1987 
Type:  Text 
Description:  290 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/71254 
Other Identifier(s):  (UMI)AAI8721642 
Date Available in IDEALS:  20141216 
Date Deposited:  1987 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois