Optimal average approximations for functions mapping in quasi-Banach spaces

Abstract

In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the ℓp-spaces for 0<p<1, and the question arose whether such a significant result might hold in some other non-Banach spaces. In this article we completely settle the problem by proving that the fundamental theorem of calculus breaks down in the context of any non-locally convex quasi-Banach space. Our approach introduces the tool of Riemann-integral averages of continuous functions, and uses it to bring out to light the differences in behavior of their approximates in the lack of local convexity. As a by-product of our work we solve a problem raised in [1] on the different types of spaces of differentiable functions with values on a quasi-Banach space.