Integration and Differentiation in a Banach Space

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Gordon, Russell Arthur

• Mathematics

Identifier

(UMI)AAI8721642

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/71254