Kurzweil-Henstock type integration on Banach spaces (Q1769690)

The main result of the paper is the following theorem: Let \(X\) be a Banach space and let \(g:[0,1]\to{X}\) be a function. Let \(g_n:[0,1]\to{X},\; n=1,2,\dots,\) be Kurzweil-Henstock-Pettis (KHP) integrable functions satisfying the following conditions: the family \(\{x^*g_n:\| x^*\| \leq1, n\in{\mathbb{N}}\}\) is Kurzweil-Henstock equiintegrable on \([0,1]\); the sequence \(\langle{g_n}\rangle_n\) is pointwise bounded on \([0,1]\); for each \(x^*\; \lim_{n\to\infty}x^*g_n=x^*g\) almost everywhere. Then \(g\) is KHP-integrable on \([0,1]\) and \(x^*g_n\to x^*g\) in the Alexiewicz norm, for each \(x^*\). If the unit ball of \(X^*\) is weak\(^*\)-separable, then \(g\) is Henstock integrable. The proof follows the idea of \textit{R. F. Geitz's} proof [Proc. Am. Math. Soc. 82, 81-86 (1981; Zbl 0506.28007)] for the Pettis integral, modulo a few necessary modifications. In particular, instead of the Vitali convergence theorem for real functions a corresponding result of \textit{C. Swartz} [Real Anal. Exch. 24, No. 1, 423--426 (1998; Zbl 0943.26022)] is applied. Reviewer's remark. A similar result, based on the same idea of Geitz, has been proven also by \textit{M. Cichon} [Acta Math. Hung. 92, No. 1--2, 75--82 (2001; Zbl 1001.26003)].