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The duals of some Banach spaces
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Type
Thesis
Author
Khaing Khaing Aye
Supervisor
Tang, Wee Kee
Abstract
The Henstock-Kurzweil integral has been used to represent continuous linear functionals and orthogonally additive functionals on the Denjoy space, namely, the space of all Henstock-Kurzweil integrable functions. In this thesis, we use the Henstock-type integral to represent linear and nonlinear functionals on the space BV of functions of bounded variation, the space RF of regulated functions, and the space BVP, 1 < P < oo, of functions of bounded p-variation.
The main results are the integration-by-parts formula for the Henstock-Stieltjes integral, convergence theorems in the above-mentioned spaces equipped with norm or two-norm structure, representation theorems for linear and nonlinear functionals defined on the above-mentioned spaces, and the characterization of compact sets in BV and BVp. The major techniques we used to prove our theorems come from real analysis and functional analysis. Specifically they are the properties of the Henstock-Stieltjes integral, two-norm convergence in BV and BVp and the nonlinear integral. The Henstock-Stieltjes integral is an extension of the Henstock integral and that of Riemann-Stieltjes. For the nonlinear integral, we adopted that of P. Y. Lee with slight modification of conditions.
The representation theorems for linear functionals defined on BV have been proved by Hildebrandt (1966) using a two-norm structure in BV. We unified the approach to include nonlinear functionals. The representation theorems for linear functionals defined on a subspace of RF have been given by Tvrd'y, Milan (1996). We extended the results to the whole space BV and further to nonlinear functionals.
For further research, we could improve the representation theorems and perhaps extend the theory to Orlicz spaces.
The main results are the integration-by-parts formula for the Henstock-Stieltjes integral, convergence theorems in the above-mentioned spaces equipped with norm or two-norm structure, representation theorems for linear and nonlinear functionals defined on the above-mentioned spaces, and the characterization of compact sets in BV and BVp. The major techniques we used to prove our theorems come from real analysis and functional analysis. Specifically they are the properties of the Henstock-Stieltjes integral, two-norm convergence in BV and BVp and the nonlinear integral. The Henstock-Stieltjes integral is an extension of the Henstock integral and that of Riemann-Stieltjes. For the nonlinear integral, we adopted that of P. Y. Lee with slight modification of conditions.
The representation theorems for linear functionals defined on BV have been proved by Hildebrandt (1966) using a two-norm structure in BV. We unified the approach to include nonlinear functionals. The representation theorems for linear functionals defined on a subspace of RF have been given by Tvrd'y, Milan (1996). We extended the results to the whole space BV and further to nonlinear functionals.
For further research, we could improve the representation theorems and perhaps extend the theory to Orlicz spaces.
Date Issued
2002
Call Number
QA322.2 Kha
Date Submitted
2002