Integration in quasi-Banach spaces and the fundamental theorem of calculus

Abstract

We make a general approach to integrability and its interplay with differentiability in quasi-Banach spaces. This endeavor demands studying first the defects of Bochner and Riemann integration in the setting of p-Banach spaces when p<1. The conclusion will be that the local convexity is a necessary (and sufficient) condition of the space for the integral operator to work in the expected way. On the positive side, we obtain a criterion for Riemann integrability of quasi-Banach valued maps based on an approximation method by polynomial functions. Finally, with an eye to finding a class of functions whose integral interacts well with differentiation, we give sufficient conditions that guarantee the fulfillment of the fundamental theorem of calculus, and prove the Lebesgue differentiation theorem for the integral in the sense of Vogt.