Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

Wee Leng Ng

doi:10.53733/114

Abstract

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

Highlights

The Banach-Steinhaus theorem is an important result in the field of functional analysis
The statement of the theorem is often given in various forms, one of which states that any family of continuous linear operators between Banach spaces is uniformly bounded provided that it is bounded pointwise
Since the Denjoy space is a Sargent space, the BanachSteinhaus theorem for H[a, b] can be obtained as a consequence of that for Sargent spaces. This is a useful result for the study of integrals which are nonabsolute, such as the Henstock-Kurzweil integral, because the Denjoy space is not a Banach space, and the Banach-Steinhaus theorem for H[a, b] can be considered as an extension of the classical version of the theorem

Summary

Introduction

The Banach-Steinhaus theorem is an important result in the field of functional analysis. Given a sequence {Tn} of continuous linear operators from a Sargent space E to a normed linear space Z, if sup { Tn(x) Z : n ≥ 1} < +∞ for every x ∈ E, sup { Tn : n ≥ 1} < +∞. The Denjoy space of [a, b], or the Denjoy space when there is no ambiguity, denoted by H[a, b], is the space of all Henstock-Kurzweil integrable functions on a closed bounded interval [a, b] equipped with the Alexiewicz norm as given below.

Results

Conclusion