This paper is divided into five sections; in a first step, in Section 2 we present some preliminaries and introduce the Henstock-Stieltjes integral via a bilinear bounded operator and the Bochner integral, together with some basic properties.
Bochner Integral. Let us recall that a function s : [a,b] [right arrow] X is called simple if there is a finite sequence [mathematical expression not reproducible] of Lebesgue measurable sets such that [E.sub.m] [intersection] [E.sub.l] = [theta] for m [not equal to] l and [mathematical expression not reproducible], and in this case the Bochner integral of s is [mathematical expression not reproducible].
the Bochner integral of f: [a,b] [right arrow] X is denoted by [mathematical expression not reproducible] and is defined by
Actually it is [L.sub.N] : C ([a, b], X) [right arrow] X, and [L.sub.N] (f) exists as a Bochner integral. If c [member of] X, then
 Appendix F, The Bochner integral and vector-valued Lp-spaces, https://isem.math.kit.edu/images/f/f7/AppendixF.pdf.
Mikusinski, The Bochner integral, Academic Press, New York, 1978.
Moreover, for z [member of] [B.sub.r], from the mean value theorem for the Bochner integral
, we obtain
A measurable function x : J [right arrow] X is Bochnerintegrable ifand only if [parallel]x[parallel] is Lebegue integrable (For properties of the Bochner integral
see Yosida ).
Some Gruss type inequalities for the Bochner integral of vector-valued functions in real or complex Banach spaces are given.
Keywords and Phrases: Gruss inequality, Bochner integral, Banach spaces, Hilbert spaces.
We called it the Bochner integral of T(*) with respect to v and it is denoted by [mathematical expression not reproducible].
We note that the strong Bochner integral [mathematical expression not reproducible] may not be an operator in L.