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additive function
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(Definition)
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Definition 1 Let $f\colon V\to \R$ 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 $x,y \in V$ .
If $f$ is additive, we find that
- $f(0)=0$ . In fact $f(0)=f(0+0)=f(0)+f(0)=2f(0)$ .
- $f(nx) = nf(x)$ for $n\in \N$ . In fact $f(nx)=f(x)+\cdots +f(x) = nf(x)$ .
- $f(nx) = nf(x)$ for $n\in \Z$ . In fact $0=f(0)=f(x+(-x)) =f(x)+f(-x)$ so that $f(-x)=-f(x)$ and hence $f(-nx)=-f(nx)=-nf(x)$ .
- $f(qx) = q f(x)$ for $q\in \Q$ . 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 $\Q$ 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 $\R$ ).
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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 2289 times total.
Classification:
| AMS MSC: | 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations) |
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Pending Errata and Addenda
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