additive function

Definition 1.

Let $f\mathrm{:}V\mathrm{\to }\mathrm{R}$ be a function on a real vector space $V$ (more generally we can consider a vector space $V$ over a field $F$).We say that $f$ is additive if

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

for all $x\mathrm{,}y\mathrm{\in }V$.

If $f$ is additive, we find that

1. $f(0)=0$. In fact $f(0)=f(0+0)=f(0)+f(0)=2f(0)$.

2. $f(nx)=nf(x)$ for $n\in \mathbb{N}$. In fact $f(nx)=f(x)+\mathrm{\cdots }+f(x)=nf(x)$.

3. $f(nx)=nf(x)$ for $n\in \mathbb{Z}$. In fact $0=f(0)=f(x+(-x))=f(x)+f(-x)$ so that $f(-x)=-f(x)$ and hence $f(-nx)=-f(nx)=-nf(x)$.

4. $f(qx)=qf(x)$ for $q\in \mathbb{Q}$. In fact $qf(px/q)=f(q(px/q))=f(px)=pf(x)$ so that $f(px/q)=pf(x)/q$.

This means that $f$ is $\mathbb{Q}$ linear.Quite surprisingly it is possible to show that there exist additive functions which are not linear (for example when $V$ is a vector space over the field $ℝ$).