Licence de mathématiques fondamentales

E=RnL(E)
X E
uL(E)
u2=id .
u F G
Rnu|F=idFu|G=−idG
u, v ∈L(E)||u◦v|| ≤ ||u|| ||v|| h∈L(E)u=id+h
Xkhkid X
X
n≥2X
u F D
(e1, . . . , en)F=Ker(u−id) = vect(e1, . . . , en−1)D=Ker(u+id) = Ren
(uk)k∈Nlimkuk=u
σnR2σn((x, y)) = (x−2ny, −y)
x−ny = 0 σn
R2
n≥2XHRn
Rn
XHL(E)
χ:L(E)→Rn[X]u7→ χuRn[X]
Rn+1
X
h., .iRn||.||
|hx, yi| =||x|| ||y|| λ x =λy
y=λx λ hx, yit
hx+ty, x +tyi ≥ 0
||x+y|| =||x|| +||y|| λ
x=λy y =λx
||.|| NRn
max(x1, x2)R2
KRnf
RnK f(K)⊂K
f K
a K a0=a p ≥1ap=1
p+1 Pp
i=0 f(i)(a)f(0) =id , f(i)=
f◦... ◦f
p∈Nap∈K f(ap) = ap+εpεp→0Rn
p→+∞

(apj)j(ap)pK
β f
KRnG
Gn=GL(n, R)K u ∈Gu(K)⊂K x ∈Rn
ν(x) = sup
u∈G
||u(x)||
xRnu∈Gn7→ u(x)∈Rn
νRnGg∈G, ν(g(x)) = ν(x)
ν(x+y) = ν(x) + ν(y)λ
x=λy y =λx u0Gν(x+y) = ||u0(x+y)||
x(1), ..., x(m)Rnν(Pm
j=1 x(j)) =
Pm
j=1 ν(x(j))x(j)
u∈GFu={x∈K , u(x) = x}
FuK
{u1, ..., up}Gw=1
pPp
j=1 uj
K w α
w K
ν(1
p
p
X
j=1
uj(α)) = 1
p
p
X
j=1
ν(uj(α))
uj(α)
Gν uj(α)
ujK Fu1∩... ∩Fup6=∅
K
G