Exercices et corrections de topologie des espaces normés

p1p2R2
pi(x1, x2) = xi
OR2p1(O)p2(O)R
H=(x, y)∈R2xy = 1HR2
p1(H)p2(H)R
F p2(F)p1(F)
F E
F=E
F E x ∈E
d(x, F )=0 ⇐⇒ x∈F
E
F E x ∈E
d(x, F ) = 0 ⇐⇒ x∈F
F G E
U V
F⊂U, G ⊂V U ∩V=∅
A, B E
d(A, B) = inf
x∈A,y∈Bd(x, y)>0
U V A ⊂U B ⊂V
R
kuk∞= sup
n∈N|un|
A={ } B={ }
C={ }
D={00}E={ }
R(N)
R(N)B(N,R)
B(N,R)k·k∞R(N)
ARx
y∈A|x−y|=d(x, A)A
B(N,R)
kuk∞= supn(|un|)
B(N,R)
(an)
B(N,R)

E(an)n≥0CP|an|
a= (an)n≥0E
kak=
+∞
X
n=0|an|
k·k E
F=(a∈E
+∞
X
n=0
an= 1)
F
E F E G
E F +G
Mn(C)
R C
E=C([0 ; 1],R)k·k∞
A=f∈Ef(0) = 0 Z1
0
f(t) dt≥1
A
∀f∈A, kfk∞>1
A=(x, y)∈R2xy = 1B={0} × R
A+B
Z R
E=R[X]
N1(P) = sup
t∈[0;1]P(t)N2(P) = sup t∈[1;2]P(t)
Ω = P∈EP(0) 6= 0
N1N2
A E
Vect(A)⊂Vect A
A E Fr(A)
F E
Fr(Fr(F)) = Fr(F)
A B E

A∩B⊂A∩B
A∩B=∅=⇒A∩B=∅
A E
A
A◦
A B E
d(A, B) = d(A, B)
d(A, B) = infx∈A,y∈Bd(x, y)
A1, . . . , AnE
Sn
i=1 Ai=Sn i=1 Ai
Tn
i=1 AiTn i=1 Ai
f:E→F A ⊂E A
kfk∞,A =kfk∞,A
A E
AFr A⊂A
A A ∩Fr A=∅
M2(C)
U=M∈ M2(C)∀λ∈Sp M, |λ|= 1
R=M∈ M2(C)∃n∈N∗, Mn=I2
R U
U M2(C)
U R
p∈ {0,1, . . . , n}RpMn(K)
p
RpMn(K)
u E
x∈E
u(x) = 1
f: [0 ; 1] →[0 ; 1]
f◦f=f
x∈[0 ; 1] f(x) = x
f
f f
f: [0 ; 1] →[0 ; 1]
f◦f=f
E u E
∀x∈E, 
u(x) ≤ kxk
n∈N
vn=1
n+ 1
n
X
k=0
uk

vn◦(u−Id)
Im(u−Id) ∩Ker(u−Id) = {0}
E
Im(u−Id) ⊕Ker(u−Id) = E
E
Im(u−Id) ⊕Ker(u−Id) = E
(vn) Im(u−Id)
E
n∈N∗Onn
FnRn[X]
Rn[X]
f:R→R
∀[a;b]⊂R, f([a;b])
y∈R, f−1({y})
f
ER
PEL(E)
GLn(R)Mn(R)
A∈ Mn(R)A−λIn
E F E
F E
U V E
U∩V E
(un)n∈N(vn)n∈N
un→+∞, vn→+∞un+1 −un→0
ε > 0n0∈Nn≥n0|un+1 −un| ≤ ε
a≥un0n≥n0|un−a| ≤ ε
{un−vp|n, p ∈N}R
cos(ln n)n∈N∗[−1 ; 1]
m−ln n(m, n)∈Z×N∗R
H(R,+) {0}
a= infx∈Hx > 0
a > 0a∈H H =aZ
a= 0 HR
cos(n)n∈N[−1 ; 1]
cos(ln n)n∈N∗[−1 ; 1]

Mn(C)
Mn(C)
R(N)
R(N)
kuk1=
+∞
X
n=0|un|
R(N)
kuk∞= sup
n∈N|un|
E[0 ; +∞[
E
2:f7→ sZ+∞
0
f2(t) dt
E0f∈E f
F E x 7→ P(e−x)e−x2/2P
R[X]E0E F E
A E
A=E
AR
∀a, b ∈A, a+b
2∈A
A]inf A; sup A[
AR∗
+
∀(a, b)∈A2,√ab ∈A
A∩(R\Q) ]inf A; sup A[
E=C([0 ; 1],R)ϕ∈E Nϕ:E→R
Nϕ(f) = kfϕk∞
NϕE ϕ−1R∗
[0 ; 1]
(un)
(un)un→+∞un+1
un→1
A=um
unm>n
[1 ; +∞[
A B E
A A ∩B
f:R→R
∀x, y ∈R, f(x+y) = f(x) + f(y)
f