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Minimax Theory and its Applications 04 (2019), No. 1, 151--160
Copyright Heldermann Verlag 2019

Dual of the Class of HKr Integrable Functions

Paul Musial
Dept. of Mathematics and Computer Science, Chicago University, 9501 S. King Drive, Chicago, IL 60628, U.S.A.

Francesco Tulone
Dept. of Mathematics and Computer Science, Palermo University, Viale delle Scienze, 90128 Palermo, Italy


We define for $1 \leq r < \infty$ a norm for the class of functions which are Henstock-Kurzweil integrable in the $L^r$ sense. We then establish that the dual in this norm is isometrically isomorphic to $L^{r'}$ and is therefore a Banach space, and in the case $r=2$, a Hilbert space. Finally, we give results pertaining to convergence and weak convergence in this space.

Keywords: Lr-Henstock-Kurzweil integral, HKr-dual, HKr-norm.

MSC: 26A39

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