Nonlinear Equations - UFRJ

120 [CH. 8: CONDITION NUMBER THEORY
for u = µ(f, x)Dd(x, y) and v = µ(f, x)‖f − g‖.
In particular, if F = H d , then D = max d i and µ = µ.
This theorem appeared in the context of the Shub-Smale condition
number (8.1) in several recent papers [25, 31, 69], with larger
constants.
Proof. Let Q(t)x be a geodesic, such as in Proposition 8.22 with
Q(0)x = x and Q(1)x = y. Then,
µ(f, x) −1 ≤ µ(g, x) −1 + ‖g − f‖
Similarly,
≤
≤
≤
µ(g ◦ Q(1), y) −1 + ‖g − f‖
µ(g, y) −1 + ‖g − g ◦ Q(1)‖ + ‖g − f‖
µ(g, y) −1 + Dd(x, y) + ‖g − f‖
µ(f, x) −1 ≥ µ(g, y) −1 − Dd(x, y) − ‖g − f‖
Now we just have to multiply both inequalities by µ(f, x)µ(g, y)
and a trivial manipulation finishes the proof.