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