133: endomorphismes remarquables d`un espace

n
(|)
u∈L(E)u∈L(E)
x, y (u(x), y)=(x, u∗(y))
∗
(uv)∗=v∗u∗
tker(u) = Im(u∗)tIm(u) = ker(u)
F u ⇔tF u∗
E E∗
B u∗u
uu∗=u∗u
u∗=u u∗=−u
||f(x) = ||f∗(x)|| x
u u∗
f
1
plansstables (x, y)f(x) =
ax +by ⇒f(y) = −bx +ay

2
E u
u
u
|||u||| = 1
|||u||| 61
P||p(ei)||2
u A
v(x)∈A x −v(x)∈AtA v
A
p q
p q
E1
E=Ker(Id −u)LIm(Id −u)u
Ker(u−Id)
[0; 1]
(Ax, x)>0>0
x6= 0 S+
n, S++ n
|ϕ(x, y)|26q(x)q(y)
Q(x+y)6Q(x) + Q(y)
Sn(R)Z++
n(R)∞

u
u
uu∗=Id
u
tAA =In
u∗
SO(E)O+(E)O−(E)
SO(E)
2
SO2(R)
E
tr(u)=2cos(θ)x[x, u(x)] = sin(θ)
3

E
E
T r(u) = 1 + 2cos(θ)
y[x, y, u(y)] = sin(θ)
3
O(E)SO(E)
O(E)GL(E)SO(E)
O(E)SO(E)O−(E)
SO(E)
Z2
On, SOn(R)tM=−M
n(n−1)/2
GL(E)O(E)
GL(E)O(E)
O(E)SO(E)Id
−Id O(E)
O(E)oZ2O(E)×Z2
O(E)P O(E)
P O+(E)P SO(E) = SO(E)
O(E)
n SO(E)
SO(E
D(SO(E)) = D(O(E)) = SO(E)
3P SO(E)n>5
P SO(E) 4 E
2SO(E)S3oSO(F)f3
SO(E)
SO(E)