derivations on a ring of continuous functions
Proof. Step one. We will prove that for any . Indeed
and thus . Now from linearity of we obtain that
Step two. If is continuous and , then is continuous and obviously Moreover, if then , but . Thus we may assume that for fixed .
Let . Now we will restrict only to such maps that .
Of course both and are continous, nonnegative and . Thus
so it is enough to show that only for nonnegative and continuous functions.
Step four. Assume that is nonnegative, continuous and . Then there exists continuous such that (indeed and it is well defined, continuous map, because was nonnegative). Then we have
Now we have and thus
Now we can take any and repeat steps two, three and four to get that for any we have
which completes the proof.
|Title||derivations on a ring of continuous functions|
|Date of creation||2013-03-22 18:37:25|
|Last modified on||2013-03-22 18:37:25|
|Last modified by||joking (16130)|