Grands Théorèmes de Topologie

f:F→R
g:E→R
f
(X, d)f:X→X f
xn+1 =f(xn)x0
(X, d) Λ f:X×Λ→
X f λ ∈Λ
∀x∈X f(x, .)f λ ∈Λ
x(λ)x:λ7→ x(λ)
(Fn)n
∀n∈NFn+1 ⊂Fn∩Fn6=∅
(X, d)
(Xi)i∈IXi
Q
i∈I
Xi
B(0,1)
E E0BE0
E0BE0
∗
A⊂Y a ∈A fn:A→Z
∀n∈Nlim
x→A
x∈A
fn(x)ln
(fn)nf A
lim
x→a
x∈A
flim
n→∞ln
lim
n→∞ lim x→a
x∈A
fn(x) = lim
x→a
x∈A
lim
n→∞fn(x)

fn:X→R
∀n∈Nfn
∀x∈Xlim
n→∞fn(x) = f(x)
x→f(x)
∀n∈N∀x∈X fn(x)≤fn+1(x)
iv)∀n∈N∀x∈X fn(x)≥fn+1(x)
fn→f
(X, d) (Y, d0)
f:X→Y
(X, d) (Y, d0)
A⊂X f :A→Y
g:X→Y g/A =f
ωf=ωg
A⊂C(X, Y )∀x∈X
A(x)
C(X, Y )
C(X, Y )
C(X, R)
C(X, C)
∀f∈A f ∈A
Rp:E→R+p(x+y)≤p(x) + p(y)∀x, y ∈E
p(λx)≤λp(x)∀(x, λ)∈E×R∗
+
f:F→R
∀x∈F f(x)≤p(x)
e
f:E→R
∀x∈E, e f(x)≤p(x),e f|F=f
6=∅A∩B6=∅
∃H
ui:E→F i ∈I

∀i∈I ui
∀x∈Esup
i∈I
kui(x)kF<+∞
(ui)i∈I
u:E→F
∃ρ > 0B(0,2ρ)⊂u(B(0,1))
B(0,2ρ)⊂u(B(0,1)) B(0, ρ)⊂
u(B(0,1))
u:E→F
u−1
u:E→F
E×F
H H
∀x∈H∃!p(x)∈C d(x, C) = d(x, p(x)) kx−p(x)k= min
y∈Ckx−yk
p(x)p(x)∈C∀y∈C Re(< x−p(x), y−p(x)>≤
0
x→p(x)
C=FHp(x)
p(x)∈F x −p(x)∈F⊥
H
Φ : H→H0
x7→ y→< x, y >
ΦH H 0
Hl∈H0x∈C
∀y∈C
Re(a(x, y −x)) ≥Re(l(x−y))
x∈C
1
2a(x, x)−Re(l(x)) = min
y∈C(1
2a(y, y)−Re(l(y)))
u:E→F
u0=tu:F0→E0
l7→ u0(l) = l◦u
Hu∈L(H)
λ∈Sp(u)⇒λ∈Sp(u∗)
F⊂H⇔F⊥u∗
u⇔u∗
Im(u)⊥=Ker(u∗)
⇒Sp(u)⊂RSp(u)⊂[m, M]
M=max
kxk=1
< u(x), x > m =min
kxk=1
< u(x), x >
Sp(u)
Spres(u)

Hu:H→H
(λn)n∈{0..N+}
(µn)n∈{0..N−}N+N−N
N+N−
Sp(u)− {0}={λnn∈ {0..N+}} ∪ {µnn∈ {0..N−}}
kuk=max(λ0,−µ0)
∀λ∈Sp(u)− {0}Eλ=Ker(u−λId)∀λ6=λ0
Eλ⊥Eλ0
H=Ker u ⊕ ⊕
n∈{0..N+}
Eλn⊕ ⊕
n∈{0..N−}
Eµn
a < b f : [a, b]→F
]a, b[g: [a, b]→R+]a, b[
∀t∈]a, b[kf0(t)k≤ g0(t)
kf(a)−f(b)k≤ g(b)−g(a)
U⊂E fn:U→E
∃x0∈U(fn(x0))n
∀a∈U∃r(a)>0B(a, r(a)) ⊂U(fn)n
B(a, r(a))
∀a∈U(fn)nB(a, r(a)) g= lim f0
n
f= lim fnf U f0=g
f:U→F
Ckk≤1a∈U df(a)
f(a)f|V:V→W
Ck
E, F, G U E ×F f :U→G
Ckk≥1 (a, b)∈U f(a, b)=0
∂f(a, b)∈L(F, G)U0
(a, b)U V a E g :V→F
Ck
(x, y)∈U0, f(x, y)=0⇔x∈V y =g(x)
Rpf:U→RnCk
k≥1f(0) = 0 f0
ψ CkRn0 0
ψ◦f(x1...xn)=(x1...xp,0...0)
Rnf:U→RqCkk≥1
f(0) = 0 f0
ϕ CkRn0 0
f◦ϕf(x1...xn) = (x1...xq,0...0)
f∀(t0, x0)∈U
u(t0) = x0