Espaces de Hilbert et opérateurs compacts

◦
xp= limn(ep|xn) (Pk
i=1 xpep, xn)→Pk i=1 |xp|2
k
X
i=1
|xp|2≤lim inf 
k
X
i=1
xpep kxnk.
Pk
i=1 |xp|2≤lim inf kxnk2P|xp|2<∞x=Pxpepp
(ep, xn)→(ep, x)F={y∈H|(y, xn)→(y, x)}F
xnenF=H xn* x
(ep, en) = δn,p →0n→+∞p en*0
={x|kxk ≤ 1}
⊂w⊂
w
en*0 0 ∈wx∈xn=x+αnenαn
kxnk= 1 αn=−(x, en) + p(x, en)2− kxk2+ 1 (xn, ep)=(x, ep)n>p
xn* x x ∈w w =
en*0emk+mkenk*0mk
mkm nk
menk*0
em6= 0 mk
E xn* x (xn)E0ϕn:`7→ `(xn)
ϕnkxnk kxnk→∞ `
sup |ϕn(`)|=∞ϕn(`) = `(xn)→`(x)
en
en*0 0
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Tn∈ K(E, F )T∈ L(E, F )kTn−Tk → 0T(BE)
ε > 0NkTN−Tk< ε TN(BE)TN(BE)
ε
B(TNxi, ε)T x ∈T(BE)i TNx∈B(TNxi, ε)
kT x −T xik≤k(T−TN)xk+kTNx−TNxik+k(TN−T)xik ≤ 3ε.

T x ∈B(T xi,3ε)T(BE)⊂ ∪B(T xi,3ε)T(BE)
F T
ε > 0T(BE)
ε B(T xi, ε)Fε= Vect(xi)PFFε
Tε=PF◦T Tεx∈BEi
kT x −T xik< ε T xi=PFT xi
kT x −Tεxk=kT x −PF(T x)k≤kT x −T xik+kPF(T xi−T x)k ≤ 2ε,
kPFk ≤ 1kT−Tek ≤ 2εlimε→0Tε=T
T E
B1 0 E
(xn)nB E xn* x
B(T xn)nT x
T
Maδkn =akδkn Ma∈ L(E, F ) (an)
kMauk`2≤ kank`∞kuk`2
an→0am= (am
nn≤m)nMam
kMa−Mamk= supn≥m|an| → 0m→ ∞ Ma
Maδk= (δkn)nδk*0
Maδk`2kMaδkk=kakδkk → 0
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(T∗fp|en)=(fp|T en)kT∗fpk2=Pn|(fp|T en)|2
X
p
kT∗fpk2=X
pX n
|(fp|T en)|2=X
nX p
|(fp|T en)|2=X
n
kT enk2,
(fp)G
(˜en)H
PnkT˜enk2=PnkT∗fpk2PnkT˜enk2<∞
unH un* u
H un=Pman,memm
an,m →amn→ ∞ u=Pmamemunun−u
C n
X
m
|anm −am|2≤C.
T T un=Pman,mT emε > 0MPm≥MkT emk2≤
ε/C N n ≥M
X
m<M
|an,m −am|2≤ε
kTk2
HS
.

kT un−T uk2=
X
m
(an,m −am)T em
2
≤
X
m<M
|an,m −am|kT emk
2
+
X
m≥M
|an,m −am|kT emk
2
≤X
m<M
|an,m −am|2X
m<M
kT emk2+X
m≥M
|an,m −am|2X
m≥M
kT emk2
≤ε
kTk2
HS
kTk2
HS +Cε
C≤2ε.
T un→T u H T
T
Maan=1
1+ln nn≥1
en
T T en=Pman,mem
K(x, y) = X
m,n
am,nen(x)em(y).
K∈L2(Ω ×Ω) TkKk2
L2=Pn,m a2
nm =kTk2
HS <∞
TKp, q (TKep|eq) = ap,q TKep=T epTK
L2(Ω) TVect(en)T
T f =RK1f=RK2f K1−K2
0 = kTK1−K2kL2=kK1−K2kL2K1=K2
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(fn, f0
n)→(f, g)
C0
fn(x) = fn(0) + Zx
0
f0
n(t)dt
f(x) = f(0) + Zx
0
g(t)dt
f C1f0=g D
Id : (F, k·kC1)→(F, k·kC0)
K D B F
C0f∈Bkf0kC0≤K B
K
B
E
U¯ U

U U
x+1
2U x ∈U
U⊂
n
[
i=1
xi+1
2U.
F= Vect(xi)U⊂F+1
2U
U⊂F+1
2kU U
U F
U⊂¯ F=F
U E ⊂F E =F
F(F, k·kCα)fn
fn→f∈F C0([0,1]) F
fnCα
|(fn−f)(x)−(fn−f)(y)|
|x−y|α= lim
m
|(fn−fm)(x)−(fn−fm)(y)|
|x−y|α
≤lim inf
mkfn−fmkCα.
supx6=y
kfn−fkCα≤ kfn−fkC0+ lim inf
mkfn−fmkCα→0n→ ∞.
fn→f Cα(F, k·kCα)
kfkC0≤ kfkCα
BF,C0(0,1)
BF,C0(0,1) BF,C0(0,1) ⊂BF,Cα(0, C)
f∈BF,C0(0,1)
∀x, y ∈[0,1],|f(x)−f(y)| ≤ C|x−y|α,
BF,C0(0,1) BF,C0(0,1)
(F, k·kC0)F
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|T f(x)| ≤ ZX×Y |k(x, y)|2dη(y)1/2ZY |f(y)|2dη(y)1/2.
X
kT fkL2(X)≤ kkkL2(X×Y)kfkL2(Y).
T
L2(X, µ)T
fn* f L2(X, µ) (T fn)nT f

f= 0 fn*0T fn(x)→0x
y7→ k(x, y)∈L2(Y, η)x
|T f(x)| ≤ ZX×Y |k(x, y)|2dη(y)1/2kfnkL2(Y).
(T fn)n
L2(X, µ)
T fn→0L2(X, µ)
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