Czechoslovak Mathematical Journal, first online, pp. 1-18
The topology of the space of $\mathcal{HK}$ integrable functions in ${\mathbb R}^n$
Varayu Boonpogkrong
Received July 24, 2022. Published online May 22, 2023.
Abstract: 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.
Keywords: compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral in $\R^n$
Affiliations: Varayu Boonpogkrong, Department of Mathematics, Division of Computational Science, Faculty of Science, Prince of Songkla University, 15 Kanjanavanich Road, Hat Yai, Songkhla 90110, Thailand, e-mail: varayu.b@psu.ac.th
