poly de cours

2
3
O(E)

ER
E ϕ :E×E→R
ϕ
∀x, y ∈E, ϕ(y, x) = ϕ(x, y)
ϕ x ∈E y ∈E7→ ϕ(x, y)
ϕ y ∈E x ∈E7→ ϕ(x, y)
ϕ ϕ(x, x)>0x∈E
ϕ x = 0 ϕ(x, x)=0
R
ϕ
ϕ(x, x)=0 x= 0
−→
x .−→ y
ϕ(x, y)<x|y> (x|y)
•E=Rn
ϕ(x, y) =
n
P
k=1
xkykX
Y ϕ(x, y) = tXY
R2ϕ(x, y) = x1y2+ 2x2y2
ϕ(x, y)=2x1y1+x2y1+x1y2+ 2x2y2=tX2 1
1 2Y
A
R∗
+(X, Y )7→ tXAY Rn
•E=Mn(R)Mn(R)Rn2
<A|B>=X
i,j
ai,j bi,j
X
16i,j6n
ai,j bi,j = (t A.B),
•R[X]Rn[X]
(P, Q)7→
n
P
k=0
pkqk
+∞
P
k=0
pkqk
(P, Q)7→ Zb
a
P(t)Q(t)dt a < b

Rn[X] (P, Q)7→
n
P
i=0
P(i)Q(i)
R[X]
∞
P
n=0
P(n)Q(n)−n
•E=C([a, b])
(f, g)7→ Zb
a
fg
<f|g>=Zb
a
fgρ,
ρ[a, b]
[0,1] t7→ sin(πt)
ϕRE
x, y ∈E|ϕ(x, y)|6ϕ(x, x)1/2ϕ(y, y)1/2
ϕ x y
λ7→<λx +y|λx +y>
a1, ..., an∈R
1
2√n n
X
k=1
ak!2 6
n
X
k=1
a2
k√k.
R2R3
−−→
AB.−−→ AB =
−−→
AB
2
ERE N :E→R+
N(λx) = |λ|N(x) (λ, x)∈R×E
N(x) = 0 x= 0 x= 0 N(x) = 0
(x, y)∈E2N(x+y)6N(x) + N(y)
ϕ E x 7→ pϕ(x, x)E
N(x+y)6N(x) + N(y)N(x+y)26(N(x) + N(y))2
<x|y > 6kxkkyk

x, y ∈E
<x|y>=1
2kx+yk2−kxk2+kyk2=1 4kx+yk2− kx−yk2
k·k
x
y
<x|y>= 0 x y
E
x, y ∈E <x|y>= 0 x⊥y
F1F2<x|y>= 0 (x, y)∈
F1×F2F1⊥F2
(y1, . . . , yk)⊥<
yi|yj>= 0 (i, j)∈[[1, k]]2i6=j yi
k⊥k
(y1, . . . , yn)
F1= Vect(x1, . . . , xk)F2= Vect(y1, . . . , yp)F1⊥F2
xi⊥yj(i, j)∈[[1, k]] ×[[1, p]]
F1⊥F2F1∩F2={0}
F1⊥F2(x1,...,xk, y1,...,yp)
X⊂F X X⊥E
X
X⊥={y∈E| ∀x∈X, <x|y>= 0}.
X{v0}v⊥
0{v0}⊥E
X⊂E X⊥E

F E F ⊥⊥F
E
{0}⊥=E E⊥={0}
F⊂G G⊥⊂F⊥
v0∈E v⊥
0= (Rv0)⊥
F∩F⊥={0}
F G E
F⊥∩G⊥= (F+G)⊥
F⊂(F⊥)⊥
F⊥+G⊥⊂(F∩G)⊥
A⊂B A B
x1, . . . , xk∈E k>2
x1⊥x2kx1+x2k2=kx1k2+kx2k2
x1, . . . , xk
kx1+··· +xkk2=kx1k2+··· +kxkk2.
x, y ∈E
kx+yk2+kx−yk2= 2(kxk2+kyk)2).
A
BC
D
AC2+BD2=AB2+BC2+DC2+AD2
E=C([0,1],R)f∈Ekfk∞|f|
k.k∞E
f g