Equivalences of Riemann Integral Based on p-Norm

Ray-Ming Chen

doi:10.3390/math6090168

Abstract

In the usual Riemann integral setting, the Riemann norm or a mesh is adopted for Riemann sums. In this article, we use the p-norm to define the p-integral and show the equivalences between the Riemann integral and the p-integral. The p-norm provides an alternative approach to define the Riemann integral. Based on this norm, we also derive some other equivalences of the Riemann integral and the p-integral.

Highlights

In this article, only the bounded functions defined on a closed interval [ a, b] are considered.A function f is used to denote a bound function defined on [ a, b]
The usual settings of a Riemann integral are listed for comparison
Since the p-norm is massively used in functional analysis, by defining an alternative Riemann integral via this norm, one could further look at the typical the Riemann integral from a new aspect

Summary

Introduction

Only the bounded functions defined on a closed interval [ a, b] are considered. A function f is used to denote a bound function defined on [ a, b]. The usual settings of a Riemann integral are listed for comparison. The p-norm and the Riemann norm turn out to be equivalent, the p-norm has many merits in connecting functional analysis and integrals. Since the p-norm is massively used in functional analysis, by defining an alternative Riemann integral via this norm, one could further look at the typical the Riemann integral from a new aspect. This might further extend the Riemann integral to other territories. Defining the p-norm from the beginning and deriving all the equivalences directly gives us some insightful knowledge between all these equivalences

Background

Notations

Definitions

Theorems

Conclusions