[pdf]

a E k
ϕ:E×E→R
ϕ(x, y) = hx, yi+khx, aihy, ai
ϕ
E=C1([0 ; 1],R)f, g ∈E
ϕ(f, g) = Z1
0
f0(t)g0(t) dt+f(1)g(0) + f(0)g(1)
ϕ E
B
x, y ∈B t ∈]0 ; 1[
k(1 −t)x+tyk<1
E=C([a;b],R)
(f|g) = Zb
a
f(t)g(t) dt
F E
{0}
x1, . . . , xnE
M∈R
∀(ε1, . . . , εn)∈ {1,−1}n,
n
X
k=1
εkxk
≤M
n
X
k=1 kxkk2≤M2
E=C([0 ; 1],R)
hf, gi=Z1
0
f(x)g(x) dx
f∈E F f
∀x∈[0 ; 1], F (x) = Zx
0
f(t) dt
v E v(f) = F
v∗
∀f, g ∈E, hv(f), gi=hf, v∗(g)i
v∗◦v
F E
F⊥=¯ F⊥
e= (ei)1≤i≤nf= (fj)1≤j≤n
E
u∈ L(E)
A=
n
X
i=1
n
X
j=1
(fi|u(ej))2
A

R[X]
hP, Qi=Z1
0
P(t)Q(t) dt
A∈R[X]
∀P∈R[X], P (0) = hA, P i?
E=R[X]
ϕ(P, Q) = R1
0P(t)Q(t) dt E
θ:E→Rθ(P) = P(0)
Q P ∈E
θ(P) = ϕ(P, Q)
Qn(X) = 1
2nn!(X2−1)n(n)
n≥1Qnn]−1 ; 1[
Qn=Xn+ (X2−1)Rn(X)
Rn∈R[X]Qn(1) Qn(−1)
(P, Q)∈R[X]2
hP, Qi=Z1
−1
P(t)Q(t) dt
QnRn−1[X]
kQnk2
R[X]
hP, Qi=Z1
−1
P(t)Q(t) dt
(Pn)
Pnn
Pn
n≥1Pn+1 −XPn
Rn−2[X]
λn∈R
Pn+1 =XPn+λnPn−1
n∈N∗E=Rn[X]
h · ,· i: (P, Q)∈E27→ hP, Qi=Z+∞
0
P(t)Q(t)e−tdt
h · ,· i
F={P∈E, P (0) = 0}d(1, F )
(P0, . . . , Pn) (1, X, . . . , Xn)
Pk(0)2
F⊥(P0, . . . , Pn)
d(1, F ⊥)d(1, F )
F= (x1, . . . , xn)n≥2
∀1≤i6=j≤n, (xi|xj)<0
n−1F
x1, x2, ..., xn+2
E n ∈N∗
∀1≤i6=j≤n+ 2,(xi|xj)<0

a E α ∈R
fα:x7→ x+α(a|x)a
fα◦fβfα
fα
E=C1([0 ; 1],R)
∀f, g ∈E, hf, gi=Z1
0
f(t)g(t) dt+Z1
0
f0(t)g0(t) dt
h., .iE
V={f∈E|f(0) = f(1) = 0}W=f∈E|fC2f00 =f
V W
W
α, β ∈R
Eα,β ={f∈E|f(0) = α f(1) = β}
inf
f∈Eα,β Z1
0f(t)2+f0(t)2dt
ϕ:R[X]×R[X]→R
ϕ(P, Q) = Z+∞
0
P(t)Q(t)e−tdt
ϕR[X]
ϕ(Xp, Xq)
inf
(a,b)∈R2Z+∞
0
e−t(t2−(at +b))2dt
inf Z1
0
t2(ln t−at −b)2dt, (a, b)∈R2
(en)n∈N
E x ∈E
kxk2=
+∞
X
n=0(en|x)
2
A∈ Mn(R)
∀X∈ Mn,1(R),kAXk≤kXk
k.k
∀X∈ Mn,1(R),
tAX ≤ kXk
A∈ Mn(R)
∀X∈ Mn,1(R),kAXk≤kXk
k.k

∀X∈ Mn,1(R),
tAX ≤ kXk
X∈ Mn,1(R)AX =XtAX =X
Mn,1(R) = Ker(A−In)⊕Im(A−In)
A∈ Mn(R)
rg(tAA) = rg A

ϕ E
ϕ(x, x) = kxk2+khx, ai2
ϕ(a, a) = kak2+kkak4= (1 + k)
ϕ
1 + k > 0
1 + k > 0
k≥0ϕ(x, x)≥ kxk2
∀x∈E\ {0E}, ϕ(x, x)>0
k∈]−1 ; 0[ k=−α α ∈]0 ; 1[
ϕ(x, x) = kxk2−αhx, ai2
hx, ai2≤ kxk2kak2=kxk2
ϕ(x, x)≥ kxk2−αkxk2= (1 −α)kxk2
∀x∈E\ {0E}, ϕ(x, x)>0
ϕ
ϕ1 + k > 0
ϕ E ×E→R
f∈E
ϕ(f, f) = Z1
0
f0(t)2dt+ 2f(0)f(1)
Z1
0
f0(t) dt2 ≤Z1
0
f0(t)2dt
Z1
0
f0(t)2dt≥(f(1) −f(0))2
ϕ(f, f)≥f(1)2+f(0)2≥0
ϕ(f, f)=0 f(0) = f(1) = 0 R1
0f0(t)2dt= 0
f
k(1 −t)x+tyk ≤ (1 −t)kxk+tkyk ≤ 1
kxk= 1 kyk= 1 (1 −t)x ty
x y
f∈F⊥f[a;b]
∀ε > 0,∃P∈R[X],kf−Pk∞,[a;b]≤ε
kfk2=Zb
a
f2=Zb
a
f(f−P) + Zb
a
fP =Zb
a
f(f−P)
Zb
a
f(f−P)≤(b−a)kfk∞kf−Pk∞≤(b−a)kfk∞ε
εkfk2= 0 f= 0 F⊥⊂ {0}
F⊥={0}
n= 1
n= 2