The space of Henstock integrable functions of two variables. (English) Zbl 0662.26003

It is shown in this note that the space of Henstock integrable functions of two variables (a multidimensional extension of the Perron integral defined through limit of Riemann sums), when equipped with the Alexiewicz norm, is barrelled. A partial description of its dual is given. For Perron integrable functions, the dual of the space is given by the class of multipliers, i.e. functions whose distributional derivatives are measures. An example shows that the two-dimensional case is different, because functions exist whose distributional partial derivatives are measures and which are not multipliers for Henstock integrable functions.


26A39 Denjoy and Perron integrals, other special integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
46A08 Barrelled spaces, bornological spaces
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