If $f:V \to W$ is a continuous group homomorphism, then $f$ is linear

Question

I'm doing Problem III.1.13 in textbook Analysis I by Amann.

I'm able to show that $f(ax)=af(x)$ for all $a\in \mathbb R$ and $x\in V$.

Let $z= a+bi \in \mathbb C$. Then $f(zx) = f((a+bi)x) = f(ax+bix) = af(x) + bf(ix)$. I'm tuck at proving $f(ix)=if(x)$.

Could you please give me some hints to finish the proof? Thank you so much!

Answer 1

You cannot prove it because it is false. Conjugation from $\mathbb C$ into itself is a continuous group homomorphism, but it is not linear.