coursMT25_ch04

E n n ∈N∗
u E E
u∀x∈E, ku(x)k=kxk
u
∀(x, y)∈E2,hu(x)|u(y)i=
=⇒4hv|wi=kv+wk2− kv−wk2
4hu(x)|u(y)i=ku(x) + u(y)k2− ku(x)−u(y)k2
=ku(x+y)k2− ku(x−y)k2u
=kx+yk2− kx−yk2u
= 4 hx|yi
=⇒u u y =x
u E u
∀x∈E, ||u(x)|| =
u
u u E
u{0E}
x∈(u)⇐⇒ u(x)=0E⇐⇒ ku(x)k= 0 ⇐⇒ kxk= 0 u⇐⇒ x= 0E

u E
u
u
u E E
u E E
=⇒
B= (e1, e2, . . . , en)E u
hu(ei)|u(ej)i=hei|eji=δi, j u(e1), u(e2), . . . , u(en)E
E
=⇒B= (e1, e2, . . . , en)E u
E
x=
n
X
j=1
xjejE
u E F
E u,
u(F) =
F⊥u

u|Fu F
F u(F) = F
x∈F⊥. y ∈F y0∈F y =u(y0).
hu(x)|yi=u(x)|u(y0)
=x|y0u
= 0 x∈F⊥y0∈F
u(x)∈F⊥F⊥
u F ⊥u(F⊥) = F⊥
F E
F sF:E−→ E
x=x1+x2x1∈F x2∈F⊥
sF(x) =
dim(F) = dim(E)−1sFF
sFF E
sF= 2 pF−EpF
sF∈L(E)sF◦sF=
sF

O(E)E
EO(E)
u∈ O(E)⇐⇒ [u∈L(E)∀x∈E, ku(x)k=kxk]
O(E)GL(E)
∀(u, v)∈ O(E)× O(E),
u◦v
u−1
E
• O(E)⊂GL(E)E∈ O(E)

A n tA A =
nOn(R)
In
E n A ∈Mn(R)
A∈ On(R)
tA A =In
AtA=In
A∈GLn(R)A−1=
A E
AMn,1(R)
A E
E
⇐⇒
ai,j A bi,j tA. C1, C2, . . . , CnA
i, j tA A
pi,j =
n
X
k=1
bi,k ak,j =
n
X
k=1
ak,i ak,j =hCi|Cji
AMn,1(R)
(i, j)pi,j =δi,j tA A =In
⇐⇒
A∈Mn(R)B= (e1, e2, . . . , en)E u ∈L(E)A=B(u)