201:espaces de fonctions. Exemples et applications.

X Y
∞X
X Y C(X, Y )Y
Cb(X, Y )Y
0∞
K
X
X Y
0∞
Y Cb(X, Y )C0(X, Y )
K Y
C1

K A C0(K, R)
X∀x f ∈A f(x)6= 0 A C0(K, R)
aifi||x−ai||
R2
fRRftn= 0
f f RRft2n= 0
f f RRft2n= 0
f= 0 fR+In:= Rf(t)e−nt = 0
n f
f u
Rfu = 0 f f
Rfu = 0 f= 0
Ck(Ω)
ΩRnk Ck(Ω)
CkΩ
∂α(fg) = Pβ6αCα
β∂βf∂βg
KRnCk(K)⊂C0(K)
k+α > k0+α0Ck0,α0(K)⊂Ck,α(K)
Ω
H(Ω)
H(Ω) CΩ
H(Ω)

C
C
ΩCa∈Ωf∈H(Ω,Ω)
f(a) = a fnf
|f0(a)|61
|f0(a)|= 1 ⇒f∈Aut(Ω)
|f0a)<1| ⇒ fna
KC
K K f
αj
S2\KΩK
K
αj
Lp
p∈[1,∞)f:x→C||f||p:=
(RX|f|pdµ)1
p.Lp(X) := {f| ||f||p<∞}
||f||∞=supess(|f|) = inf {C| |f(x)|6C pp} L∞(X) :=
{f|||f||∞<∞}
||f||∞= lim
p→+∞||f||p
p∈[1,∞]. p p0
p0∈[1,∞]1
p+1 p0= 1
f g
ZX
fg 6(ZX fp)1
p(ZX
gp0)1
p0
f∈Lp(X)g∈Lq(X)fg ∈Lr(X)1
p+1 q=1 r
f g
||f+g||p6||f||p+||g||p
(Lp(X),||.||p)
f∈ L(X)p||f||p= 0 ⇔f= 0
fRg ⇔ ||f−g||p= 0 ⇔f=g Lp(X) =
Lp(X)/R ||Cl(f)||p:= ||f||p
(Lp(X),||.||p)

Lp(X)
Lp(X)
Lp(X)
Lp
Lp
P
Lp
Lp
Lpf=f+−f−
f∈L1(Rn)RRnf(x)ei<x,ξdx →0
ξ∞
f∈L1(Rn)∀ε > 0,∃δ > 0|mes(A)< δ ⇒RAf6ε
Lp(X)
µ p 6q⇒LqLp(X)
µ
f:x7→ |x|s1[−1;1] ∈Lp(R)
f:x7→ |x|s1[1;∞)∈Lp(R)
Lp
f∈L1(Rn)F(f)(ξ) = RRnf(x)ei<x,ξdx
FL10
L1⊂L∞)
FL2L2
F
L1f∗g(x) =
Rf(t)g(x−t)dt
L1×L1L1
L1×L2L2
1
p+1 q=1 r+ 1 Lp×LqLr
Lp×LqLr
ΩLp
F(f∗g) = F(f)F(g)

Hs
s>0Hs(Rn) := f∈L2(Rn)|Rn
R((1 + x2)s)|F(f)|2dx < ∞
(f, g) = RRn((1 + x2)s)|F(¯ f)F(g)dx
Hs||.||HSC∞
0
s
s L2
C∞
0
Hs(Rd)⊂L∞
06s < d/2 : Hs(Rd)⊂Lp1/p > 1/2−s/d
s > d/2 + k:Hs(Rd)⊂C(k)
→0
s(Rd)Hs
Rns>0
iut−∆u= 0
u(0, .) = u0∈Hs
u
∀ϕ∈S∀t∈RRRniut−∆uϕdx = 0
S(t)u0:= e−it∆u0:= ¯ F(e−it|ξ|2F(u0)(ξ)) S(t)u0∈C0(R, Hs(Rn))
||S(T)u0||HS=||u0||HsS(t)
s > n
2)
iut−∆u=P(u, ¯u)∈L2
u(0, .) = u0∈Hs
0 0 u[−T, T ]
u∈C0([−T, T ], Hs)u A(u) := S(t)u0−iRt
0S(t−
σ)P(u)(σ)dσ =u
C0([−T, T ], Hs)
T
Lp
T T ∗