additive function

Definition 1.

Let f:VR be a functionMathworldPlanetmath on a real vector space V (more generally we can consider a vector spaceMathworldPlanetmath V over a field F). We say that f is additive if


for all x,yV.

If f is additive, we find that

  1. 1.

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

  2. 2.

    f(nx)=nf(x) for n. In fact f(nx)=f(x)++f(x)=nf(x).

  3. 3.

    f(nx)=nf(x) for n. 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. 4.

    f(qx)=qf(x) for 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 linear. Quite surprisingly it is possible to show that there exist additive functionsMathworldPlanetmath which are not linear (for example when V is a vector space over the field ).

Title additive function
Canonical name AdditiveFunction
Date of creation 2013-03-22 16:17:31
Last modified on 2013-03-22 16:17:31
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 9
Author paolini (1187)
Entry type Definition
Classification msc 15A04
Related topic LinearFunctional