Compacité

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E, F f :E−→ F. Gr(f) = {(x, y)∈E×F y =
f(x)}
f Gr(f)E×F
f(E)F
f(E)
A E f :A−→ A∀x, y ∈A, x 6=y⇒d(f(x), f(y)) <
d(x, y)
f a
(xn)A xn+1 =f(xn)a
C E f :C−→ C, 1f
fnx7−→ a
n+1−1 nf(x)a∈C
A E f :A−→ F F
f−1:f(A)−→ A
A f−1
A E f :A−→ A∀x, y ∈A, d(f(x), f(y)) ≥d(x, y)
a∈A(an)a0=a an+1 =f(an)a
(an)
a, b ∈A d(f(a), f(b)) = d(a, b)
f(A) = A
A E (ak)A
A
E(Kn)E K =
T
n
Kn
K6=∅
U K n Kn⊂U
δ(K) = lim
n→∞ δ(Kn)δ
E, F f :E−→ F(Kn)
E f(∩
nKn) = T
n
f(Kn)

A E (Oi)i∈IA
r > 0A r Oi
E=B(N,R) = {u= (un)}Ekuk=
∞
P
n=0
|un|
2n
A={u∈E∀n∈N,0≤un≤1}
E=C([0,2π]) k.k2n∈Nfn(x) = cos(nx)
kfn−fpk2n, p ∈N
B(0,1)
k.kR2A⊂R2
A
k.k=k.k∞
k.k
∗
E K E f K K
1f
E
F a ∈E\F
b∈Fka−bk=d(a, F )
c∈Ekck=1=d(c, F )
E

d(xn, a)d(xnk)` d(`, a) = d
(f(xnk)) f(`)d(f(`), a) = d ` =a
C fn1−1
nxn∈C fn(xn) = xn
(xn)f
`= lim
n→∞ δ(Kn)xn, yn∈Knd(xn, yn) = δ(Kn)
xn> x yn> y ε > 0 (B(x, ε)∩Kn) (B(y, ε)∩Kn)
B(x, ε)∩K B(y, ε)∩K
δ(K)≥`−2ε
Ui,n ={x∈E B(x, 1/n)⊂Oi}Ui,n E
⇒r= min(1/n)
(un)A un= (un
k) (unp0)
(unp0
0)u0∈[0,1] (unp1) (unp1 0, unp1 1)
(u0, u1)∈[0,1]2(unpk)kA(u0, u1, . . . )
a∈K n ≥1fnx7−→ 1
na+1−1 nf(x)fn
1−1
nK xnx(xn)
f(x) = x