there exist additive functions which are not linear

Example 1.

There exists a function $f\colon\mathbb{R}\to\mathbb{R}$ which is additive but not linear.

Proof.

Let $V$ be the infinite dimensional vector space $\mathbb{R}$ over the field $\mathbb{Q}$ . Since $1$ and $\sqrt{2}$ are two independent vectors in $V$ , we can extend the set $\{1,\sqrt{2}\}$ to a basis $E$ of $V$ (notice that here the axiom of choice is used).

Now we consider a linear function $f\colon V\to\mathbb{R}$ such that $f(1)=1$ while $f(e)=0$ for all $e\in E\setminus\{1\}$ . This function is $\mathbb{Q}$ -linear (i.e. it is additive on $\mathbb{R}$ ) but it is not $\mathbb{R}$ -linear because $f(\sqrt{2})=0\neq\sqrt{2}f(1)$ . ∎

Title	there exist additive functions which are not linear
Canonical name	ThereExistAdditiveFunctionsWhichAreNotLinear
Date of creation	2013-03-22 16:17:50
Last modified on	2013-03-22 16:17:50
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	5
Author	paolini (1187)
Entry type	Example
Classification	msc 15A04