Nonlinear Equations - UFRJ

[SEC. 5.2: METRIC STRUCTURE ON ROOT SPACE 59
Proposition 5.9. Let F be a fewspace. Let 〈u, w〉 x = ω x (u, Jw) be
the (possibly degenerate) Hermitian product associated to ω. Then,
〈u, w〉 x = 1 2
∫
F x
(Df(x)u)Df(x)w
K(x, x)
dF x (5.1)
where dF x =
1
(2π) dim Fx e−‖f‖2 dλ(f) is the zero-average, unit variance
Gaussian probability distribution on F x .
Proof. Let
P x = I −
K(·, x)K(·, x)∗
K(x, x)
be the orthogonal projection F → F x . We can write the left-handside
as:
〈u, w〉 x = 〈P xDK(·, x)u, P x DK(·, x)w〉
K(x, x)
For the right-hand-side, note that
Df(x)u = 〈f(·), DK(·, x)u〉 = 〈f(·), P x DK(·, x)u〉.
Let U =
1
‖K(·,x)‖ P xDK(·, x)u and W =
1
‖K(·,x)‖ P xDK(·, x)w.
Both U and W belong to F x . The right-hand-side is
∫
1 (Df(x)u)Df(x)w
2 F x
‖K(x, x)‖ 2 dF x = 1 ∫
〈f, U〉〈f, W〉 dF x
2 F x
= 1 ∫
2 〈U, W〉 1
2π |z|2 e −|z|2 /2 dz
which is equal to the left-hand-side.
= 〈U, W〉
For further reference, we state that:
Lemma 5.10. The metric coefficients g ij associated to the (possibly
degenerate) inner product above are
(
1
g ij (x) = K ij (x, x) − K )
i·(x, x)K·j (x, x)
K(x, x)
K(x, x)
C