Fréchet derivative is unique
The Fréchet derivative is unique.
Proof. Assume that both and in satisfy the condition for the Fréchet derivative (http://planetmath.org/derivative2) at the point . To prove that they are equal we will show that for all the operator norm is not greater than . By the definition of limit there exists a positive such that for all
holds. This gives
Now we have
thus as we wanted to show.
|Title||Fréchet derivative is unique|
|Date of creation||2013-03-22 16:08:35|
|Last modified on||2013-03-22 16:08:35|
|Last modified by||Mathprof (13753)|