TITLE: Non linear transformation satisfying $T(x+y)=T(x)+T(y)$
QUESTION [5 upvotes]: Given V a vector space with vectors and scalars $\mathbb{C}$, does there exists a non linear transformation $T:V\rightarrow V$ such that $T(x+y)=T(x)+T(y)$ for all $x,y\in V$?
I think such a transformation will be 'like' one that satisfies Cauchy's functional equation $f(x+y)=f(x)+f(y)$ without any other conditions, but other than that, I have no idea.

REPLY [2 votes]: The function $T \colon V \to V$ is a homomorphism of abelian groups or equivalently $\mathbb{Z}$-modules.  If we use Zorn's Lemma we can realise $V$ as a freely generated module over $\mathbb{Z}$ with generators $X$ say.  Then $X$ is uncountable.  Picking $X$ is like picking a basis of a complex vector space.  You can define every  $T$ by taking any function $X \to V$ and extending it to  the whole of $V$ to be a homomorphism like you extend any map of a basis to be a linear map.  Unfortunately I don't think there is any constructive way of finding $X$ so all this shows is there are lots of maps $T$.
Might these all be linear ?  No as you can arrange them not to be.  As $X$ is uncountable you can choose $\dim(V) + 1$ distinct generators from $X$.  These must be linearly dependent so arrange $T$ to not be linear on them and then extend it to the rest of $X$ arbitrarily.
EDIT: Previous comment on automorphisms of $\mathbb{C}$ removed as not really relevant.