COMPACT OPERATORS AND INTEGRAL EQUATIONS IN THE HK SPACE

The space HK of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces HK(X). In this paper, on each Frechet space HK(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 HK(X) space. It is known that every control-convergent sequence in the HK space always belongs to a HK( X) space for some X. We illustrate how to apply results for Frechet spaces HK(X) to control-convergent sequences in the HK space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.