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Vector Space Real Number Relation Theory Field Additive Function There Exist Additive Functions Which Are Not Linear

1	Vector Space
1	Field
1	Real Number
1	Relation Theory
2	Additive Function
14	There Exist Additive Functions Which Are Not Linear

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Additive Function

Definition 1 Let $f\colon V\to \mathbb{R}$ be a function on a real vector space (more generally we can consider a vector space over a field ). We say that is additive if

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

for all $x,y \in V$ .

If is additive, we find that

. In fact .
for $n\in \mathbb{N}$ . In fact $f(nx)=f(x)+\cdots +f(x) = nf(x)$ .
for $n\in \mathbb{Z}$ . In fact so that and hence .
for $q\in \mathbb{Q}$ . In fact so that .

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