213 - Espaces de Hilbert. Bases hilbertiennes. Exemples - IMJ-PRG

K=R C
HKH
K
`2(C) = {(xn)n∈CN|Pn|xn|2<+∞}
hx, yi=Pnxnyn
(X, µ)L2(X, µ) = {f∈CX|
RX|f(x)|2dµ(x)<+∞} hf, gi=RXf(x)g(x)dµ(x)
(E, k.k)
x, y ∈Ekx+yk2+kx−yk2= 2kxk2+kyk2
x, y ∈Ekx+yk2+
kx−yk2= 2kxk2+kyk2
(H, h., .i)C
H x ∈H p ∈Ckxpk= miny∈Ckx−yk
x C pCx
C C Re(hx−pC(x), y−
pC(x)i)≤0y∈C pC

F H x ∈H pF(x)
p∈H p ∈F x −p∈F⊥pF:H→F
H
H=F⊕F⊥F F ⊥={0}
H
A B H A B
f∈H∗supa∈Af(a)<
infb∈Bf(b)
H H0
φ:H→H0, y 7→ (x7→ hx, yi)
u H y ∈H
z∈H x ∈Hhu(x), yi=hx, ziu∗(x) = z
u∗H u
H f H g
H∀x, y ∈H, hf(x), yi=hx, g(y)if
u
u H H
H= ker u∗⊕⊥imu
T H
kTk ≤ 11
n+1 Pn
k=0 Tk(x)→n→+∞p(x)p
ker Id −T

xn∈HNx∈H
xn* x ∀y∈H, hxn, yi→hx, yi
xn* x kxnk→kxk
xn→x
H(ei)i∈I
H
∀i6=j∈I, hei, eji= 0
∀i∈I, keik= 1
H= vectei
H(Vi)i∈I
vectVi=H
δi,j l2
H
(en)n∈NH
(en)n∈N
x∈H, s =Pnhx, enien
x∈H, kxk2=Pn|hx, eni|2
{en|n∈N}⊥={0}
∆ : H→`2(C), x 7→ (hx, eni)n∈N

H
H
H T
H T
IRρ
α > 0RIeα|x|ρ(x)dx < ∞
L2(I, ρ)
L2(en)
P1/n2
1
a±2iπx