The topology of the space of $\mathcal{H}\mathcal{K}$ integrable functions in ${\mathbb{R}^n}$

Abstract

It is known that there is no natural Banach norm on the space $\mathcal{H}\mathcal{K}$ of n-dimensional Henstock-Kurzweil integrable functions on [a, b]. We show that the $\mathcal{H}\mathcal{K}$ space is the uncountable union of Fréchet spaces $\mathcal{H}\mathcal{K}(X)$. On each $\mathcal{H}\mathcal{K}(X)$ space, an F-norm ‖·‖^X is defined. A ‖·‖^X-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F-norm is also defined for a ‖·‖^X continuous linear operator. Hence, many important results in functional analysis hold for the $\mathcal{H}\mathcal{K}(X)$ space. It is well-known that every control-convergent sequence in the $\mathcal{H}\mathcal{K}$ space always belongs to a $\mathcal{H}\mathcal{K}(X)$ space. Hence, results in functional analysis can be applied to the $\mathcal{H}\mathcal{K}$ 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.