203 - Utilisation de la notion de compacité. - IMJ-PRG

K
K
(X, d)
X
(K, d)
(Oi)iK λ > 0
K λ Oi
(K, d)ε > 0
K(x1,...,xn)xi
ε K
R
R C

R C
E F
T:E→F T
HT:H → H
HT
R
R
f:K→
K K
(x, y)K d(f(x), f (y)) < d(x, y)a
x0∈K(fn(x0))na
f:K→X
K X K X f
f
S1
R
K X f :K→
X K f

K(fn)n
C0(K, R) (fn)
f
(fn)
(Pn) [0,1] P0= 0
Pn+1 =Pn+x−P2
n
2x7→ √x
(K, d)
S⊂C0(K, R)∀f, g ∈S, max(f, g)
min(f, g)∈S
S⊂C0(K, R)∀x, y ∈K, x 6=
y, ∃f∈S:f(x)6=f(y)
K
S
C0(K, R)S S
C(X, R)
f∈C0([0,1],R)R1
0f(t)tndt n ∈N
f= 0
E
fn* f fn(x)→f(x)∀x∈E
K
f:K→K
µ f µ(f−1A) = µ(A)
A
X
A B X f :X→
[0,1] A B
X A
X f :A→[0,1] f X
[0,1]

K
(Ui)i∈IX
ϕi:X→[0,1] supp(ϕi)⊂UiPiϕi= 1 K
X(Y, d)
A ⊂ C0(X, Y )
Ax∈X∀ε > 0,∃V∈ Vx:∀f∈
A, y ∈V=⇒d(f(x), f(y)) < ε
A
AX x ∈X
{f(x)|f∈ A} Y
ΩRdω
ω⊂ΩF
Lp(Ω) 1 ≤p < +∞
∀ε > 0,∃δ > 0 : ∀f∈ F,∀khk< δ, kf(.+h)−fkLp(ω)< ε
F|ωLp(ω)
y0=f(t, y)f:U→RnUR×Rn
(t0, y0)∈U r0>0, T > 0
C. = [t0−T, t0+T]×B(y0, r0)⊂U T sup(t,y)∈Ckf(t, y)k< r0
(t0, y0) [t0−T, t0+T]
ΩCF
H(Ω) Ω F
FΩ
n A
n+ 1 A

E X, Y ⊂E
H E f =α f X Y
ε > 0f(x)< α−ε∀x∈X f(y)> α+ε∀y∈Y
E
K, F ⊂E K
F E
K F
KRn
f:K→K
ERC
E f C C
f(C)f
E f :E→E
C > 0∀(λ, x)∈[0,1] ×
E, λf(x) = x=⇒ kxk ≤ C f