A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions

Received Jan 15, 2018; Revised Mar 21, 2018; Accepted Apr 5, 2018

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1. Introduction

Henstock in [1] defines a Riemann type integral which is equivalent to Denjoy integral and more general than the Lebesgue integral, called the Henstock integral. Cao in [2] extends the Henstock integral for vector-valued functions and provides some basic properties such as the Saks-Henstock Lemma.

Schwabik in [3] considers a bilinear form, defines a Stieltjes type integral, and performs a study about it including [4]; following his ideas we give integration by parts theorem involving a bilinear operator and, through it, we prove a representation theorem for the space of Henstock vector-valued functions.

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. In Section 3 we provide two useful kinds of integration by parts theorems, one of them in terms of the Bochner integral and the other using Henstock-Stieltjes integral; the representation theorem is proved in Section 4 which, if we consider real-valued functions, provides an alternative proof of the representation theorem proved by Alexiewicz (Theorem 1 in [5]).

2. Preliminaries

Throughout this paper $X$, $Y$, and $Z$ will denote three Banach spaces, ${‖·‖}_X$, ${‖·‖}_Y$, and ${‖·‖}_Z$, which will denote their respective norms, $X^{\ast }$ the dual of $X$, $B:X\times Y\to Z$ a bounded bilinear operator fixed, and $\left[a,b\right]$ a closed finite interval of the real line with the usual topology and the Lebesgue measure, which we denote by $\mu$. For a function $f:\left[a,b\right]\to \mathbb{R}$ we denote the Lebesgue integral of $f$ on a measurable $E\subset \left[a,b\right]$, when it exists, by $\left(L\right){\int }_E‍f$.

Definition 1.

$B:X\times Y\to Z$ is a bounded bilinear operator if $B$ is linear in each variable and there exists $M>\mathrm{0}$ such that ${‖B\left(x\right)‖}_{}$

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