Espaces de Hilbert

VR
V
V(., .)V:V×V→R
∀v, w ∈V(v, w)V= (w, v)V
∀v∈V(v, v)V≥0
(v, v)V= 0 ⇐⇒ v= 0
(., .)VV x ∈
V7→ kxkV=p(v, v)V
∀v, w ∈V, |(v, w)V| ≤ kvkVkwkV.
v, w ∈
V→(v, w)V

l2={ui∈R;Piu2
i<∞}
u, v ∈l2< u, v >=Piuivi
|uivi| ≤ u2
i/2 + v2 i/2
<·,·>
l2kukl2=pPiu2
i
l2
H K
H u ∈Hu∈K
u K PKu
kPKu−ukH= inf
w∈Kkw−ukH
PKu
PKu∈K∀w∈K, (PKu−u, w) = 0
u∈K PKu=u
PKH K kPKkL(H,K)=
1K6={0}
u∈H ϕ :w∈H7→ (u, w)∈R
HR
H0H0
H ϕ ∈H0=
L(H, R)u∈H
∀w∈H, ϕ(w) = (u, w)H.
kukH=kϕkH0H0
H ϕ u H

A V V
v∈V anA
H
H(en)H n
kenk= 1 (en, em) = 0 n6=m
(en)H
H en
u H un= (u, en)PnunenPnu2
n
HR
u=X
n
unen,kuk2
H=X
n
u2
n