Integration and differentiation in a Banach space
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. 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.
- Authors:
- Publication Date:
- Research Org.:
- Illinois Univ., Urbana (USA)
- OSTI Identifier:
- 6448706
- Resource Type:
- Thesis/Dissertation
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BANACH SPACE; FUNCTIONS; RIEMANN FUNCTION; DIFFERENTIAL EQUATIONS; INTEGRAL EQUATIONS; VECTORS; EQUATIONS; MATHEMATICAL SPACE; SPACE; TENSORS; 657000* - Theoretical & Mathematical Physics
Citation Formats
Gordon, R A. Integration and differentiation in a Banach space. United States: N. p., 1987.
Web.
Gordon, R A. Integration and differentiation in a Banach space. United States.
Gordon, R A. 1987.
"Integration and differentiation in a Banach space". United States.
@article{osti_6448706,
title = {Integration and differentiation in a Banach space},
author = {Gordon, R A},
abstractNote = {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. 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.},
doi = {},
url = {https://www.osti.gov/biblio/6448706},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Thu Jan 01 00:00:00 EST 1987},
month = {Thu Jan 01 00:00:00 EST 1987}
}