Exercise 3.A.9

Question

Give an example of a function $\varphi:\mathbf{C}	o \mathbf{C}$ such that $$\varphi(w+z)=\varphi(w)+\varphi(z)$$for all $w,z\in \mathbf{C}$ but $f$ is not linear. (Here $\mathbf{C}$ is thought of as a complex vector space.)

Moon Flower · Accepted Answer

Consider the function $\mathrm{Im}$ defined by $\mathrm{Im}(z)=\frac{\mathrm{conj}(z)-z}{2i}$ where 
$$
\mathrm{conj}(z)=\begin{cases}
\frac{|z|^2}{z} & 	ext{if }z
eq 0 \
0 & 	ext{otherwise}
\end{cases}
$$
We can verify that it is additive; let $z=z_{1}+z_{2}i$ and $w=w_{1}+w_{2}i$, then it follows that $\mathrm{Im}(z+w)=\mathrm{Im}((z_{1}+w_{1})+(z_{2}+w_{2})i)=z_{2}+w_{2}=\mathrm{Im}(z)+\mathrm{Im}(w)$.

Additionally, it does not satisfy homogeneity: $\mathrm{Im}(cz)$ for some complex numbers $c$ and $z$ is strictly real, while $c\mathrm{Im}(z)$ can be complex. Any $c$ that has an imaginary part will not satisfy homogeneity.

Thus the function is not linear. $\blacksquare$

Exercise 3.A.9

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Exercise 3.A.9

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