Toward an optimal theory of integration for quasi-Banach-space-valued functions

TL;DR

This paper introduces a new integral for functions valued in quasi-Banach spaces, addressing limitations of classical integrals in non-locally convex settings and providing a fundamental theorem of calculus.

Contribution

It develops a novel integration approach based on the galb of the space, applicable to all known spaces of galbs, improving upon previous methods.

Findings

Defines a suitable integral for quasi-Banach spaces

Addresses deficiencies of Bochner and Riemann integrals in non-locally convex spaces

Establishes a fundamental theorem of calculus for the new integral

Abstract

We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for $p$-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is not always possible to choose an optimal $p$ to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.