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Keywords:
compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral in $\mathbb{R} ^n$
compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral in $\mathbb{R} ^n$
Summary:
It is known that there is no natural Banach norm on the space $\mathcal {HK}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathcal {HK}$ space is the uncountable union of Fréchet spaces $\mathcal {HK}(X)$. On each $\mathcal {HK}(X)$ space, an $F$-norm $\|{\cdot }\|^X$ is defined. A $\|{\cdot }\|^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a $\|{\cdot }\|^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathcal {HK}(X)$ space. It is well-known that every control-convergent sequence in the $\mathcal {HK}$ space always belongs to a $\mathcal {HK}(X)$ space. Hence, results in functional analysis can be applied to the $\mathcal {HK}$ space. Compact linear operators and the existence of solutions to integral equations are also given. The results for the one-dimensional case have been discussed in V. Boonpogkrong (2022). Proofs of many results for the $n$-dimensional and the one-dimensional cases are similar.
References:
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