Abstract

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 (xα)(x_{\alpha}) is an infinite linearly independent subset of a Fréchet space XX and if CC is the convex cone generated by (xα),(x_{\alpha}), then CC 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:Rn→Rf:{\mathbb R}^n \to {\mathbb R} is an order monotone function, then ff is Baire measurable, but in general, ff is not universally measurable

Author information

Contact details are reproduced from the original publication and may be historical.

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.

zbigniew@uic.edu

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Suggested citation

P. Fischer, Z. Slodkowski. “Hamel Bases, Convexity and Analytic Sets in Fréchet Spaces.” Journal of Convex Analysis 24 (2017), No. 3, 999–1014. https://doi.org/10.68381/jca24060

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.