Nonlinear Equations - UFRJ

86 [CH. 7: NEWTON ITERATION
Proposition 7.2, only if. Assume that the series (7.2) converges uniformly
for ‖x − x 0 ‖ < ρ. Without loss of generality assume that
E = F and Df(x 0 ) = I.
We claim that
lim sup
k≥2
sup
‖u‖=1
‖ 1 k! Dk f(x 0 )u k ‖ 1/k ≤ ρ −1 .
Indeed, assume that there is δ > 0 and infinitely many pairs
(k i , u i ) with ‖u i ‖ = 1 and
In that case,
‖ 1 k! Dk f(x 0 )u k ‖ 1/k > ρ −1 (1 + δ).
‖ 1 ( ) k ρ
k! Dk f(x 0 ) √ u ‖ > √ 1 + δ k
1 + δ
infinitely many times, and hence (7.2) does not converge uniformly
on B(x 0 , ρ).
Now, we can apply Lemma 7.3 to obtain:
lim sup
k≥2
‖ 1 k! Dk f(x 0 )‖ 1/(k−1) ≤ e lim sup
k≥2
sup
‖u‖=1
≤ e lim
k→∞ ρ−(1+1/(k−1))
= eρ −1
and therefore ‖ 1 k! Dk f(x 0 )‖ 1/(k−1) is bounded.
‖ 1 k! Dk f(x 0 )u k ‖ 1
k−1
Exercise 7.1. Show the polarization formula for Hermitian product:
〈u, v〉 = 1 ∑
ɛ‖u + ɛv‖ 2
4
ɛ 4 =1
Explain why this is different from the one in Lemma 7.3.
Exercise 7.2. If one drops the uniform convergence hypothesis in the
definition of analytic functions, what happens to Proposition 7.2?