Publication: The duals of some Banach spaces
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 <i>BV</i> of functions of bounded variation, the space <i>RF</i> of regulated functions, and the space <i>BV</i><sub>P</sub>, 1 < P < oo, of functions of bounded <i>p</i>-variation.<br><br>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 <i>BV</i> and <i>BV</i><sub>p</sub>. 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 <i>BV</i> and <i>BV</i><sub>p</sub> 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.<br><br>The representation theorems for linear functionals defined on <i>BV</i> have been proved by Hildebrandt (1966) using a two-norm structure in <i>BV</i>. We unified the approach to include nonlinear functionals. The representation theorems for linear functionals defined on a subspace of <i>RF</i> have been given by <i>Tvrd'y, Milan</i> (1996). We extended the results to the whole space <i>BV</i> and further to nonlinear functionals.<br><br>For further research, we could improve the representation theorems and perhaps extend the theory to Orlicz spaces.