Henstock - Kurzweil Integral for Banach Valued Function

Abstract

In this paper, we have studied the Henstock - Kurzweil integral which is a generalized Riemann integral means. Hen-stock - Kurzweil integral is the natural extension of Riemann integral. We defined Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation which is an extension of real valued Henstock - Kurzweil integral with respect to an increasing function. We investigated elementary properties of the Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation. We proved the convergence theorems and Saks - Henstock lemma of the Henstock - Kurzweil integral of Banach valued functions with respect to a function of bounded vari-ation. Equi-integrability with respect to Banach space valued function is defined and equi-integrable theorem of Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation is proved. Finally Bochner Henstock - Kurzweil integral of Banach valued function with respect to a function of bounded variation is defined and the relation between Bochner Henstock - Kurzweil integral and Henstock - Kurzweil integral is exhibited.