Corrigé

E=R2[X]ϕ:E→R
ϕ(P, Q) = P(0)Q(0) + P(1)Q(1) + P(2)Q(2)
ϕ
ϕ
ϕ
P∈E=R2[X]ϕ(P, P ) = P(0)2+P(1)2+P(2)2≥0
ϕ
ϕ
P∈E=R2[X]ϕ(P, P )=0 P(0)2+P(1)2+P(2)2= 0
P(0) = P(1) = P(2) = 0
P≤2ϕ
ϕ
ϕ
q ϕ q
q
E
rg(q) = 3 sgn(q) = (3,0)
F={P∈E / P (0) = 0}F E
F
P∈E=R2[X]a0, a1, a2P=a0+a1X+a2X2
P∈F⇐⇒ P(0) = 0 ⇐⇒ a0+a1.0 + a2.0 = 0 ⇐⇒ a0= 0 ⇐⇒ P=a1X+a2X2⇐⇒ P∈V ect(X, X2)
F=V ect(X, X2) (X, X2)
(X, X2)F
F ϕ
(X, X2)F
P1=X
kXk
ϕ(X, X) = 0.0+1.1+2.2 = 5 kXk=pϕ(X, X) = √5
P1=1
√5X

P0
2=X2−ϕ(X2, P1)P1
ϕ(X2, P1) = ϕµX2,1
√5X¶=1 √5ϕ(X2, X) = 1 √5¡0.0 + 1.1+22.2¢=9 √5
P0
2=X2−9
√5.1 √5X=X2−9
5X P 0
2
ϕ(P0
2, P 0 2) = 0 + µ1−9
5¶2
+µ4−9
5.2¶2
=16
25 +4 25 =20 25 =4 5=⇒ kP0
2k=2
√5
P2=√5
2µX2−1 5X¶
P1P2F ϕ
F⊥
dim(F) + dim(F⊥) = dim(E) dim(F⊥) = 3 −2 = 1
dim(F⊥) = 1
P≤2X X2
½ϕ(P, X) = 0
ϕ(P, X2) = 0 ⇐⇒ ½0 + P(1) + 2P(2) = 0 0 + P(1) + 4P(2) = 0 ⇐⇒ ½P(1) = 0 P(2) = 0
P2
1 2
F⊥=V ect ((X−1)(X−2))
E < x|y > E
f E
∀x, y ∈E, < f(x)|y >=−< x|f(y)>
(f) = ( (f))⊥
x∈(f)y∈(f)f(y) = 0 ∃z∈E / x =f(z)
< x|y >=< f(z)|y >=−< z|f(y)>=−< z|0>= 0
(f) (f)
(f)⊂( (f))⊥
dim( (f)) = dim(E)−dim( (f))
dim ³( (f))⊥´= dim(E)−dim( (f)) (f) ( (f))⊥
((f)⊂( (f))⊥
dim ( (f)) = dim ( (f))⊥
(f) = ( (f))⊥

E= (f) + (f)
(f) = ( (f))⊥(f)∩(f) = {0}
x∈(f)∩(f)< x|x >= 0 x(f)
x(f)x= 0
(f) + (f)⊂E(f) (f)E
dim( (f) + (f)) = dim( (f)) + dim( (f)) −dim( (f)∩(f)) = dim( (f)) +
dim( (f)) = dim(E)
E= (f) + (f)
(f) = (f2)
⊂x∈(f)f(x) = 0
f2(x) = f(f(x)) = f(0) = 0 x∈(f2)
⊃x∈(f2)f2(x) = 0
f(x)∈(f)∩(f)f(x) = 0 x∈(f)
(f) = (f2)
x∈E < f(x)|x >= 0
x∈E < f(x)|x >=−< x|f(x)>=−< f(x)|x >
f
∀x∈E, < f(x)|x >= 0
f
λ f
x f(x) = λx
0 =< f(x)|x >=< λx|x >=λ < x|x >=λkxk2
xkxk 6= 0 λ= 0
Sp(f) = {0}
E3B= (e1, e2, e3)
E f B
A=
0−p−q
p0−r
q r 0
f(e1) = me1+pe2+qe3(e1, e2, e3)E
< f(e1)|e1>=m , < f(e1)|e2>=p , < f(e1)|e3>=q
< f(e1)|e1>= 0 m= 0
A

f(e2) = ae1+be2+ce3
< f(e2)|e1>=a , < f(e2)|e2>=b , < f(e2)|e3>=c
< f(e2)|e2>= 0 b= 0
a=< f(e2)|e1>=−< e2|f(e1)>=−p a =−p
r=c
f(e3) = xe1+ye2+ze3
< f(e3)|e1>=x , < f(e3)|e2>=y , < f(e3)|e3>=z
< f(e3)|e3>= 0 z= 0
x=< f(e3)|e1>=−< e3|f(e1)>=−q x =−q
y=< f(e3)|e2>=−< e3|f(e2)>=−r y =−r
fB
A=
0−p−q
p0−r
q r 0
f◦f E
x, y ∈E < f ◦f(x)|y >=< x|f◦f(y)>
< f ◦f(x)|y >=−< f(x)|f(y)>=−(−< x|f◦f(y)>) =< x|f◦f(y)>
f◦f E
f◦f
λ f ◦f x f ◦f(x) = λx
< x|f◦f(x)>=
< x lambdax >=λkxk2
−< f(x)|f(x)>=−kf(x)k2
λkxk2≤0λ≤0
Sp(f◦f)⊂R−
f◦f
f◦f f
B0f◦f
D=
λ10
λ2
λn
λ1, . . . , λnf◦f
D
f◦f f f ◦f
u f ◦f λ
(u, f(u))
u f ◦f u f ◦f(u) = λu
f
< u|f(u)>= 0
u f(u)u(u, f(u))