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Journal of Convex Analysis 24 (2017), No. 3, 999--1014
Copyright Heldermann Verlag 2017

Hamel Bases, Convexity and Analytic Sets in Fréchet Spaces

Pal Fischer
Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ont. N1G 2W1, Canada

Zbigniew Slodkowski
Department of Mathematics, University of Illinois, Chicago, IL 60607-7045, U.S.A.


It is shown that a Hamel basis over the field of reals of an infinite dimensional linear Polish space can not be an analytic set. Furthermore, if $(x_{\alpha})$ is an infinite linearly independent subset of a Fr\'echet space $X$ and if $C$ is the convex cone generated by $(x_{\alpha}),$ then $C$ is not a closed set. In particular, the convex cone generated by a Hamel basis in such a space can not be closed.The notion of convex and midpoint convex functions extended to the case when the domain of the functions is a connected open set, and analytic graph theorems are given for these functions. It is shown also that if $f:{\mathbb R}^n \to {\mathbb R}$ is an order monotone function, then $f$ is Baire measurable, but in general, $f$ is not universally measurable.

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