Riemann integrability under weaker forms of continuity in infinite dimensional spaces

TL;DR

This paper explores how classical properties of Riemann integrability and continuity break down in infinite dimensional spaces, especially in quasi Banach spaces, and discusses the implications for vector integration.

Contribution

It analyzes the failure of classical integrability properties in infinite dimensional spaces with weaker topologies, extending the understanding beyond Banach spaces.

Findings

Classical Riemann integrability properties do not hold in infinite dimensional spaces.

Weaker topologies impose severe restrictions on integrability in quasi Banach spaces.

The paper discusses recent developments in vector integration theory.

Abstract

In classical analysis, the relationship between continuity and Riemann integrability is an intimate one: a continuous function on a closed and bounded interval is always Riemann integrable whereas a Riemann integrable function is continuous almost everywhere. In the setting of functions taking values in infinite dimensional spaces that include quasi Banach spaces, one encounters certain curious situations involving the breakdown of the above stated phenomena, besides the failure of the fundamental theorem of calculus and the non-existence of primitives for continuous functions! While some of these properties can surely be salvaged within the class of Banach spaces, it turns out that certain important properties involving vector integration that include Riemann integration no longer hold in an infinite dimensional setting. This will be seen to be the case, for example, in situations when it is required to integrate functions which are continuous with respect to certain well known ( generally compatible) linear topologies on X (resp. its dual) weaker than the norm topology. As we shall see in Section 3(a), such a requirement imposes rather severe restrictions on the space in question. The present paper is devoted to a discussion of these issues which will be examined in the setting of Banach and Frechet spaces on the one hand and of quasi Banach spaces on the other. The paper concludes with a brief (but non-technical) description on recent developments and the current of art involving various other aspects of vector integration