Importance of the homogeneity assumption in definition of linear map

Let $V$ and $W$ be vector spaces over field $F$. A function $f: V \rightarrow W$ is said to be linear if for any two vectors $x$ and $y$ in $V$ and any scalar $\alpha\in F$, the following two conditions are satisfied:

1. $f(x + y) = f(x) + f(y)$


2. $f(\alpha x) = \alpha f(x)$

Let $F$ be a field of real numbers. Is it possible to construct $f$ such that the first condition is satified but not the second one?

Answer

Yes, it is possible to construct examples of $f$ that are additive but not homogeneous. However, the construction won't be explicit and depends on the axiom of choice. See here for details.

If you want an explicit example of an additive map that is not homogeneous over a different field, let $\mathbb{F} = \mathbb{C}$ and consider the map $\varphi \colon \mathbb{C} \rightarrow \mathbb{C}$ given by complex conjugation:

$$ \varphi(z) = \varphi(x + iy) = \overline{z} = x - iy. $$

The map $\varphi$ is additive and $\mathbb{R}$-homogeneous but not $\mathbb{C}$-homogeneous as $\varphi(az) = \overline{az} = \overline{a}\overline{z} = \overline{a}\varphi(z)$.