Espaces vectoriels - Page Personnelle de Jérôme Von Buhren

F G
E F ∪G E F ⊂G G ⊂F
F=n(un)∈RN| ∀n∈N, un+2 =nun+1 +uno.
FRN
F=n(un)∈RN| ∃T∈N∗,∀n∈N, un+T=uno.
FRN
C F (R,R)
E=f−g|(f, g)∈C2.
EF(R,R)
RN
n(un)∈RN|(un)Po
P
F(R,R)
{f:R→R|fP}
P0
u= (1,...,1) ∈Kn
H={(x1, x2, . . . , xn)∈Kn|x1+x2+··· +xn= 0}.
HKn=H⊕Vect(u)
a0, . . . , an∈R
F={f:R→R|f(a0) = ··· =f(an)=0}
F(R,R)
F=f∈C1(R,R)|f(0) = f0(0) = 0
C1(R,R)
F=f∈C0([−1,1] ,C)Z1
−1
f(t)t= 0
C0([−1,1] ,C)

F=f∈C0([0, π],R)|f(0) = f(π/2) = f(π)
C0([0, π],R)
A∈R[X]\ {0}
F={P∈R[X]|A P }.
FR[X]
FR[X]
FR[X]
G={P∈R[X]| ∃Q∈R[X], P = (1 −X)Q(X2)}.
F G R[X]
R[X] = F⊕G
E F, G, H
E F G H
F∩G={0}(F+G)∩H={0}
F1, F2
F1F2
(cos,sin,exp,Id) F(R,R)
(t7→ 1, t 7→ Arctant, t 7→ Arctan(1/t))
F(R∗,R)F(R∗
+,R)
(a, b, c)∈R3R R
x7→ sin(x+a), x 7→ sin(x+b), x 7→ sin(x+c)
P1, . . . , PnC[X]
(P1, . . . , Pn)
((X−a)−1)a∈RK(X)
F(R,R)
(i) (t7→ |t−a|)a∈R,(ii) (t7→ taebt)a∈R+, b∈R,(iii) (t7→ cos(at))a∈R+,
(iv) (t7→ sin(at))a∈R∗
+,(v) (t7→ sinn(t))n∈N∗.
(t7→ eλt)λ∈C
F(R,C)C
ϕa:R[X]→RP7→ P(a) (ϕa)a∈R
L(R[X],R)

(v1, . . . , vn)ER
∀k∈J1, n −1K, wk=vk+vk+1 wn=vn+v1.
(w1, . . . , wn)
(u1, . . . , un)E(α1, . . . , αn)∈Kn
v=
n
X
i=1
αiui.
α∈Kn(u1+v, . . . , un+v)
R Q
(1,√2,√3)
(ln(p))p∈PP
F=(P∈R[X]P(X+ 1) =
+∞
X
n=0
P(n)(X)
n!).
FR[X]
Xk∈F k ∈N
F(R,R)
Vect(x7→ cos(nx))n∈N= Vect(x7→ cosn(x))n∈N.
a0, . . . , an∈K
∀i∈J0, nK, Li(X) =
n
Y
j=0
j6=iX−aj
ai−aj∈K[X].
(L0, . . . , Ln)Kn[X]
0 = x0< x1<··· < xn= 1
F={f∈F([0,1],R)| ∀k∈J0, n −1K, f|[xk,xk+1]}.
FF([0,1],R)
F
p∈N
f:C[X]→C[X], f :P7→ (1 −pX)P+X2P0.
f
f
ϕ:C∞(R)→C∞(R)f7→ f−f0
ϕ
ϕ
:K(X)→K[X]

B∈K[X]q(P)r(P)
P B
q r
r
(f, g)∈L(E)2
g◦f◦g=g f ◦g◦f=f.
Im(f) Ker(g)E
f(Im(g)) = Im(f)
f E
Im(f)∩Ker(f) = {0} ⇔ Ker(f) = Ker(f2)
E= Im(f) + Ker(f)⇔Im(f) = Im(f2)
ϕ∈L(R[X],R)
∀P∈R[X], ϕ((X−a)P)=0.
λ∈Rϕ(P) = λP (a)P∈R[X]
u E F
E
u−1(u(F)) Ker(u)
u(u−1(F)) Im(u)
F u(u−1(F)) = u−1(u(F))
E f ∈L(E)
f2−3f+ 2Id = 0.
f f
E= Ker(f−Id) ⊕Ker(f−2Id)
f∈L(E)
∀x∈E, ∃λ∈K, f(x) = λx.
f
E, F f ∈L(E, F )A, B
E
f(A)⊂f(B)⇔A+ Ker(f)⊂B+ Ker(f).
f:E→K
u∈E\Ker(f)E= Ker(f)⊕Vect(u)
E=F1⊕ ··· ⊕ Fr
∀i∈J1, rK,Fi={u∈L(E)|Im(u)⊂Fi}.
FiL(E)
L(E) = F1⊕ ··· ⊕ Fr
E, F, G u ∈L(E, F )
v∈L(F, G)w=v◦u w
u , v Im(u)⊕Ker(v) = F.

E=F1⊕···⊕Fr
ui∈L(ui)i∈J1, rK
∃!u∈L(E),∀i∈J1, rK, u|Fi=ui.
Ker(u) = Ker(u1)⊕ ··· ⊕ Ker(ur),Im(u) = Im(u1)⊕ ··· ⊕ Im(ur).
∆ : R[X]→R[X], P 7→ P(X+ 1) −P(X)
∆
Ker(∆) Im(∆)
∆n
P∈Rn−1[X]
n
X
k=0
(−1)n−kn
kP(X+k) = 0.
D= Vect(1,0,0) P
x+y+z= 0 R3
D P
D P
(p, q)∈L(E)2
p◦q=p q ◦p=q
p q
p, q ∈L(E)2F
λ∈Kλp + (1 −λ)q
F
p, q ∈L(E)
p+q p ◦q=q◦p= 0
p+q
p, q ∈L(E)
(p, q)L(E)
p, q ∈L(E)
p◦q E
p◦q
p∈L(E)λ∈K
λ6=−1 Id + λp
p, q ∈L(E)p◦q= 0
r=p+q−q◦q E
r