PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Medium Entry average rating: No information on entry rating
additive function (Definition)
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

  1. $f(0)=0$ . In fact $f(0)=f(0+0)=f(0)+f(0)=2f(0)$ .
  2. $f(nx) = nf(x)$ for $n\in \N$ . In fact $f(nx)=f(x)+\cdots +f(x) = nf(x)$ .
  3. $f(nx) = nf(x)$ for $n\in \Z$ . In fact $0=f(0)=f(x+(-x)) =f(x)+f(-x)$ so that $f(-x)=-f(x)$ and hence $f(-nx)=-f(nx)=-nf(x)$ .
  4. $f(qx) = q f(x)$ 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 $\R$ ).




"additive function" is owned by .
(view preamble | get metadata)

View style:

See Also: linear functional

Log in to rate this entry.
(view current ratings)

Cross-references: there exist additive functions which are not linear, additive, field, vector space, real, function
There is 1 reference to this entry.

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 MSC15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)