Espaces compacts et connexes

R2
A={(x, y)∈R2|x2−y2−2xy ≤1},
B={(x, y)∈R2|x2+y2+ 3exy ≤1},
C={(x, y)∈R2|2 cos(x) + 2 sin(x)>3}.
Mn(R)k(mij )k= max |mij |
On⊂Mn(R)n×n
(mij )∈O(i, j)|mij | ≤ 1
O
Mn(R)
E(Kn)nE
Kn+1 ⊂Knn K =TnKnO K ⊂O
n Kn⊂O
(xn)n∈EN∀n xn∈Knxn→x∈E x ∈K
6⊂
KnK0
K
E
S={v∈E| kvk= 1}E B ={v∈E| kvk ≤ 1}
B S
S(vn)nB
ϕ:N→N(kvϕ(n)k)n
(vn)nB
X= [0,1]N[0,1] N
[0,1] u v ∈X d∞(u, v) = supk∈N|u(k)−v(k)|δnX
δn(k) = 1k=n
0
d∞(δn, δm)n m
(δn)n∈XN
X
Esup
E
X= [0,1]N[0,1] N
[0,1] u v ∈X
d(u, v) =
+∞
X
k=0
|u(k)−v(k)|
2k.
(un)nX ϕk:
N→Np(un(0))n(un(p))nϕ0◦ · · · ◦ ϕp
ψ(n) = ϕ0◦ · · · ◦ ϕn(n)n
k∈N(ψ(n))n≥k(ϕ0◦ · · · ◦ ϕk(n))n
(uψ(n)(k))nk
(uψ(n))n∈XNX d
(X, d)

f: [a, b]→Rmn(f) = Rb
af(t)tndt n ∈N
Rb
af(t)P(t)dt= 0 P
f f = 0
f f = 0
f:R+→RLf :R∗
+→R
(Lf)(s) = Z+∞
0
f(t)e−stdt.
s0>0 (Lf)(s0+n)=0 n∈N
R1
0us0+n−1f(−ln u)du= 0 n
f= 0
C⊂R2
C
P∈C C \ {P}
f:C→R
f(C) [a, b]
R2R
R2R
f:R2→Ra∈RA=f−1({a})
BR2f(B)f(cB)
f(cA)A
Mn(K)
GLn(K)⊂Mn(K)
GLn(K)
GLn(R)
GLn(R)Mn(R)
M /∈GLn(C) 0n
GLn(C)
M∈GLn(C)InGLn(C)
C C0⊂[0,1]
I= [a, b]R
T(I) = a, a +1
3(b−a)ta+2 3(b−a), b.
A=FIkT(A) = ST(Ik)
Cn=Tn([0,1]) C=TnCn
X={0,1,2}N∗π:X→[0,1]
π((ak)k) =
∞
X
k=1
ak
3k.
C0=π(Y)Y={0,2}N∗⊂X π Y
F:R→[0,1[
C1C2C3
3·Cn=Cn−1∪(2 + Cn−1)
π(Y)⊂Cnn
F(3n−1·Cn)⊂C1C⊂π(Y)
C=C0
C
CRC
Y Y
Leb(C) = 0 Leb

x < y C z ∈]x, y[z /∈C
C
C
f: [0,1] →RC
|f(x)−f(y)| ≤ C|x−y|x y ∈[0,1] Lip(f)C
Lip(f) = sup
x6=y
|f(x)−f(y)|
|x−y|.
E[0,1] R0
Lip E
f∈Ekfk∞≤Lip(f)
(fn)n(E, Lip)
(fn)nf: [0,1] →R
(fn)nE f : [0,1] →R
Lip(fp−fq)≤L p q Lip(fp−f)≤L f ∈E
(E, Lip)