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additive function (Definition)
Definition 1   Let $ f\colon V\to \mathbb{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
$\displaystyle f(x+y)= f(x) + f(y) $
for all $ x,y \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)+\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) = q f(x)$ for $ q\in \mathbb{Q}$. In fact $ q f(px/q) = f(q(px/q))=f(px) = p f(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 $ \mathbb{R}$).



"additive function" is owned by paolini.
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See Also: linear functional


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there exist additive functions which are not linear (Example) by paolini
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Cross-references: there exist additive functions which are not linear, additive, field, vector space, real, function
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This is version 6 of additive function, born on 2006-10-01, modified 2006-10-06.
Object id is 8409, canonical name is AdditiveFunction2.
Accessed 1140 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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