Théorème d`Ascoli

(E, d) (F, δ)
AC(E, F )
A
∀x∈E, ∀ε > 0,∃η > 0/∀f∈A, ∀y∈E, (d(x, y)≤η)⇒(δ(f(x), f(y)) < ε)
x∈E A(x) = {f(x), f ∈A}
•AC(E, F )A
ε > 0
f1, . . . , fpA
A⊂
p
[
i=1
B(fi, ε)
f A ε fi
x E f(x)ε fi(x)
A(x)⊂
p
[
i=1
B(fi(x), ε)
ε > 0A(x)
F A(x)
fiE
∃ηi>0/∀x, y ∈E, (d(x, y)< ηi)⇒(δ(fi(x), fi(y)) < ε)
η= min
i=1...p ηi>0x, y E d(x, y)< η
f A i max
z∈Eδ(f(z), fi(z)) < ε η
δ(fi(x), fi(y)) < ε
δ(f(x), f(y)) ≤δ(f(x), fi(x)) + δ(fi(x), fi(y)) + δ(fi(y), f(y)) <3ε
x y d(x, y)< η f ∈A
A
•AC(E, F )A(x)
x
C(E, F )A
A
(fp)pA E
D={ak, k ∈N}E

X=Y
k∈N
A(ak)
A(x)x∈E
F X
f∈A(f(ak))kX(fp)p
hp= (fp(ak))k∈X(hp)p∈XNE
(hϕ(p))ph= (h(k))kpk
X
∀k, pk(hϕ(p)) = fϕ(p)(ak)−→ h(k)∈F
(fϕ(p))pE
(fϕ(p))p∈AN
E F
E
E
(fϕ(p))pE(fϕ(p))p
C(E, F )A