Espaces vectoriel normés

−−−−−−−−−−−−−−−−−−
(E, N)
o
B(x, r)E
B(x, r)E
A E
o
AA E A
Ao
A=A
A A E A
A A =A
o
B(x, r) = B(x, r)
o
\
B(x, r)= o
B(x, r)
(E, N)R C
A A E o
A
A⊂E x ∈E A A
E CA
A(A)
(A) = A∩CA
A
E=RA1= [a, b]A2=]a, b[A3=ZA4=Q
E=R2A1=o
B(x, r)A2=B(x, r)
A E
A(A)⊂A
A(A)∩A=∅
A E
a∈A b ∈CA
[a, b]⊂E A
(E, N)F E
F F E E
o
FF E E
F E
(i)F
(ii) 0 ∈o
F
(iii)F=E
F E
F E

E(un)n∈NPun
EC
N1: (un)7→
+∞
X
n=0 |un|
N2: (un)7→ sup
n∈N|un|
N3: (un)7→
+∞
X
n=0
|un|
2n
E E
S E C(un)
+∞
X
n=0
un
S E N1E N2
E
E f E E
f
f E
B(0,1) f
f E
f E
(xn)E(f(xn))
(E, N)ϕ E
a∈Eker ϕ(a)
E
a E = ker ϕ⊕.a p1ker ϕ .a
p2.a ker ϕ
ϕker ϕ E p1p2
[a, b]RE=C([a, b],R) [a, b]
∀f∈E, kfk= sup
t∈[a,b]|f(t)|
ϕ E
∀f∈E, ∀x∈[a, b], f(x)>0 =⇒ϕ(f)>0
ϕ
ϕ E ∀f∈E, |ϕ(f)|6ϕ(|f|)
e∀x∈[a, b], e(x) = 1
ϕ E
e
Sa= (an)n≥1SB SCV
PCV (an)n≥1Pan
S0

SB N((an)) = sup
n≥1|an|
(an)∈ S0D((an)) = (a0
n)n≥1∀n∈N∗, a0 n=an+1 −an
DS0PCV
(bn)∈PCV rbnnPanrbn=∞
X
k=n+1
bk
an(an) = D−1((bn)) Pbn
D|||D|||
p∈Nt(p)= (t(p)
n)t(p) n= 1 1 ≤n≤p
t(p)
n= 0 p < n
N∞(t(p))N∞(D−1(t(p))) D−1
f:x−→ x2R
g:x−→ 1
x]0,+∞[
E=R C f
C1IRE
f0I f I
E=C([a, b],R) [a, b]k k∞
ΦE∀f∈E, Φ(f) = Zb
a
f(t)dt
Φ|||Φ|||
c∈[a, b]ϕcE
∀f∈E, ϕc(f) = f(c)
ϕc|||ϕc|||
n∈N∗ψnE ψn=b−a
n
n−1
X
k=0
ϕa+kb−a
n
∀f∈E, ψn(f) = b−a
n
n−1
X
k=0
fa+kb−a
n
f∈Elim
n→+∞ψn(f)
E∗∗ E E
ϕN −→ sup
f∈E, kfk∞≤1|ϕ(f)|E∗∗
lim
n→+∞N(ψn−Φ) = 0
Φ (ψn) (E∗∗, N )
(gn)gn(x) = cos λ(x−a)λ
k gn(a+kb−a
n) = 1
A∈Mn( ) (Ak)B
Bker(A−In) (A−In)
P B =P(A)P(1) 6= 0
SB a = (an)n≥1
N∞((an)) = sup
n≥1|an|
•SF
(an)n≥1SF ∃n0∈N∗,∀n≥n0, an= 0

•SCV
•S0
SF SB S0S0SB
µ a = (an)n≥1
µ(a) = b= (bn)n≥1n
∀n∈N∗, bn=a1+a2+... +an
n
µ∈L(SB)
µ SB SB |kµk|
SB µ µ(a) = a
∀a∈SF, µ(a)∈S0
ΛSCV C
a∈S0
(u(k))k∈N∗SF a
(µ(u(k)))k∈N∗µ(a)∈S0
u L ∈Cµ(u)L
l2l2
l2
l2={(un)∈RN,Pu2
n}
(un) (vn)l2Punvn
l2
(un)−→ k(un)k2=v
u
u
t∞
X
n=0
u2
nl2
n∈NTnn(uk)k≥0
(vk)vk=ukk≤n vk= 0 k > n
Tnl2
u∈l2,lim
n→∞ Tn(u) = u
a= (an)∈l2u= (un)l2Panun
ΦauΦa(u) = ∞
X
n=0
anun
l2|||Φa|||
Ψl2R
(bn)n≥0∈l2Ψ = Φb
bnΨ ∆i= (δi,n)n≥0M∈R+
∀n,
n
X
k=0
b2
k≤M.v
u
u
t
n
X
k=0
b2
kPb2
n
x1=1
5(2 sin x1+ cos x2)
x2=1
5(cos x1+ 3 sin x2)(x1, x2)R2
(a, b) 10−10
Mn(R)
GLn(R)On(R)Mn(R)
E=C([a, b],C) [a, b]
C([a, b],C)k k∞
C([a, b],C)k k1