Czechoslovak Mathematical Journal | 2021
Compact operators and integral equations in the $\\mathcal{HK}$ space
Abstract
The space $${\\cal H}{\\cal K}$$\n of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\\cal H}{\\cal K}$$\n (X). In this paper, on each Frechet space $${\\cal H}{\\cal K}$$\n (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\\cal H}{\\cal K}$$\n (X) space. It is known that every control-convergent sequence in the $${\\cal H}{\\cal K}$$\n space always belongs to a $${\\cal H}{\\cal K}$$\n (X) space for some X. We illustrate how to apply results for Frechet spaces $${\\cal H}{\\cal K}$$\n (X) to control-convergent sequences in the $${\\cal H}{\\cal K}$$\n space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.