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On functions whose graph is a Hamel basis

Author: Krzysztof Plotka
Journal: Proc. Amer. Math. Soc. 131 (2003), 1031-1041
MSC (2000): Primary 15A03, 54C40; Secondary 26A21, 54C30
Published electronically: August 28, 2002
MathSciNet review: 1948092
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Abstract: We say that a function $h \colon \mathbb{R}\to \mathbb{R} $ is a Hamel function ( $h \in {\rm HF}$) if $h$, considered as a subset of $\mathbb{R} ^2$, is a Hamel basis for $\mathbb{R} ^2$. We prove that every function from $\mathbb{R} $ into $\mathbb{R} $ can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all $f_1,f_2 \in \mathbb{R} ^{\mathbb{R} }$ there is a $g\in\mathbb{R} ^{\mathbb{R} }$ such that $g+f_1,g+f_2\in \mathrm{HF}$. We show that this fails for infinitely many functions.

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Additional Information

Krzysztof Plotka
Affiliation: Department of Mathematics, University of Scranton, Scranton, Pennsylvania 18510

Keywords: Hamel basis, additive functions, Hamel functions.
Received by editor(s): September 13, 2001
Received by editor(s) in revised form: November 2, 2001
Published electronically: August 28, 2002
Additional Notes: Most of this work was done when the author was working on his Ph.D. at West Virginia University. The author wishes to thank his advisor, Professor K. Ciesielski, for many helpful conversations.
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society