additive function
Definition 1.
Let f:V→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∈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∈ℕ. In fact f(nx)=f(x)+⋯+f(x)=nf(x).
-
3.
f(nx)=nf(x) for n∈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)=qf(x) for q∈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 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).
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 |