THE TOPOLOGY OF THE SPACE OF HK INTEGRABLE FUNCTIONS IN Rn

It is known that there is no natural Banach norm on the space HK of n-dimensional Henstock-Kurzweil integrable functions on [a, b]. We show that the HK space is the uncountable union of Fr & eacute;chet spaces HK(X). On each HK(X) space, an F-norm ||<middle dot> ||(X) is defined. A k<middle dot>kX-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F-norm is also defined for a ||<middle dot>||(X)-continuous linear operator. Hence, many important results in functional analysis hold for the HK(X) space. It is wellknown that every control-convergent sequence in the HK space always belongs to a HK(X) space. Hence, results in functional analysis can be applied to the 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.