# Functional equation using Hamel basis

Do you know a functional equation, the solution of which is $f:\mathbb{R}\mapsto \mathbb{R}$, with $f$ a function that is not expressed in function of an additive function and that requires the use of Hamel bases (basis of $\mathbb{R}$ over $\mathbb{Q}$) ?

• I don't think so. In order for the Hamel basis to play a role, we need that the function can be solved on each one-dimensional $\Bbb Q$-subspace independently. But these subspaces are defined by their additive structure, so in some way any example will be a modification of the additive function equation. (It's just that I don't know how to formalize "in some way", and so I cannot prove this gut feeling claim) – Hagen von Eitzen May 5 '17 at 21:01
• Can you clarify what you mean by "not expressed in function of an additive function and that requires the use of Hamel bases"? You can define lots of functions by taking a Hamel basis and doing something, for example forming an indicator function on a $\mathbb{Q}$-subspace of $\mathbb{R}$. – Joppy May 5 '17 at 22:31
• For example the solution of the functional equation $f(x+y)=f(x)f(y)$ is exp($g(x)$) with $g(x)$ an additive function (that is $g(x)$ is a solution of the Cauchy's functional equation). The general form of $g(x)$ is obtained by means of Hamel basis. Therefore, for what I'm searching, the solution exp($g(x)$) is ok because it uses Hamel bases but is not ok since $g(x)$ is additive. – ketherok May 6 '17 at 6:30