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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.

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    At the end of the first paragraph, I think you mean for all $x, y \in V$.2012-10-11
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    edited, thanks! :)2012-10-11
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    Hint: With $\mathbb C$ regarded as a one-dimensional vector space over itself, does $T(x) = \text{Re}(x)$ define a linear transformation or a nonlinear transformation from $\mathbb C$ to $\mathbb C$?2012-10-11
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    Thanks for the hint (although it might as well be an answer) :D2012-10-11
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    Without the assumption that $T(kx) = kT(x)$ for all $k \in \mathbb{C}$, you still have $T(kx) = kT(x)$ for all $k \in \mathbb{Q}$ (fun exercise).2012-10-11

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