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. 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.
- Research Organization:
- Illinois Univ., Urbana (USA)
- OSTI ID:
- 6448706
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
Similar Records
Regularity of solutions to an inhomogeneous differential equation in Banach space
Morse theory on banach manifolds