This is an announcement for the paper "Optimal average approximations for functions mapping in quasi-Banach spaces" by Fernando Albiac and Jose L Ansorena.
Abstract: In 1994, M. M. Popov [On integrability in F-spaces, Studia Math. no 3, 205-220] 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 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 [F. Albiac and J.L. Ansorena, Lipschitz maps and primitives for continuous functions in quasi-Banach space, Nonlinear Anal. 75 (2012), no. 16, 6108-6119] on the different types of spaces of differentiable functions with values on a quasi-Banach space.
Archive classification: math.FA
Mathematics Subject Classification: 46A16, 46G05
Remarks: 14 pages
Submitted from: joseluis.ansorena@unirioja.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1308.6127
or