1. Normes. Topologie d`un espace vectoriel normé. 1.1. Normes

K=R C
E K E ν :E→R+
(N1)ν(x) = 0 =⇒x= 0E
(N2)∀x∈E , ∀λ∈K , ν(λx) = |λ|ν(x)
(N3)∀x∈E, ∀y∈E , ν(x+y)≤ν(x) + ν(y)
(E, ν)E ν E
||.|| x∈E||x|| x
R C
Kn, n ∈N∗x= (x1, ..., xn)∈Kn
||x||eucl =¡n
P
j=1
|xj|2¢1/2;||x||∞=max
1≤j≤n|xj|ν+(x) =
n
P
j=1
|xj|
f∈E=C([0,1],C)7→ ||f||∞:= sup
0≤t≤1
|f(t)|E
X F ||f||∞:= sup
t∈X
||f(t)||
B(X, F )X F
E K ||.|| E
||0E|| = 0
¯
¯||x|| − ||y||¯ ¯≤ ||x−y|| ,∀x, y ∈E
E6={0E}E
(E, ||.||)
d: (x, y)∈E×E7→ d(x, y) := ||x−y|| E
∀x, y, z ∈E d(x+z, y +z) = d(x, y)||.||
E
Rn
RnC([0,1],C)
E
(E, ||.||)d K
x7→ ||x|| (E, d)
(x, y)7→ x+y(E, d)×(E, d)
(λ, x)7→ λx K ×E
(λ, x)∈K×E7→ λx
τa:x7→ x+a a E hλ:x7→ λx λ
K E E
(λ, x)7→ λx
E=K
A E ∀x, y ∈A , ∀t∈[0,1] : tx + (1 −t)y∈A

E
E E
ν ν
(F, ν|F)
F E F
(Ej,||.||j)j= 1, .., n E =
n
Q
1
Ej
ν∞:x= (x1, ..., xn)∈E7→ max
1≤j≤n||xj||jν:x7→ ¡n
P
j=1
|xj|2¢1/2ν+:x7→
n
P
j=1
|xj|
0E
M≥0||u(x)≤M||x|| ,∀x∈E
ν1ν26={0E}α > 0
β > 0∀x∈E:α ν1(x)≤ν2(x)≤β ν1(x)
E
K E K F E
E=F=C1([0,1],R)|f||∞=sup{|f(t)|, t ∈[0,1]}u:f∈E7→ f0
(E, ||.||∞)
f K E Ker f
E
E E
E/H = 1
ker f
E F
L(E, F )L(E)L(E, E)
E E0:= L(E, K)
|||.||| :u∈ L(E, F )7→ |||u||| = sup
kxk≤1
ku(x)k L(E, F )
|kuk|
|kuk| ku(x)k ≤ Mkxk,∀x∈E
|kuk| = sup
kxk=1
ku(x)k= sup
x6=0
ku(x)k
kxk

L(E, F )
Ejj= 1, ..., n F K f :E1×... ×En−→ F
f
f
∃M≥0,∀(x1, ..., xn)∈Qn
j=1 Ej||f(x1, .., xn)|| ≤ M||x1||...||xn||
L(E1, ..., En;F)E1×... ×EnF
E1, ...EnF K
L(E1, ..., En;F)kfk= sup
max kxjk≤1
kf(x1, ..., xn)k
kfkMkf(x1, ..., xn)k ≤ Mkx1k...kxnk(x1, ..., xn)∈E1×...×En
L(E1, ..., En;F)
n∈N∗
KnK E
n∈N∗{e1, ..., en}KnN1Knβ > 0
(x1, ..., xn)∈Kn
max
1≤j≤n|xj| ≤ β . N1(
n
X
j=1
xjej).
E, E1, .., Ep
E1×... ×Ep
B(0E,1)
(xn)n∈Nsn=x0+... +xn(xn)n
xn((xn)n,(sn)n)
xn(sn)n(sn)n
P∞
n=0 xnxn
xnxn→0n+∞
xnynλ∈K
xn+λynP∞
n=0 xn+λP∞ n=0 yn

Pxn
lim
n→+∞¡sup
p≥0
|| Pn+p
k=nxn||¢= 0
xnP∞
n=0 ||xn||
X F un
B(X, F )X F sup
t∈X
||un(t)||
PunX
P∞
n=0 an
σNaσ(n)
P∞
n=0 an=P∞ n=0 aσ(n)
L(E)E
E, F, G K u ∈ L(E, F )v∈ L(F, G)v◦u∈ L(E, G)
|||v◦u||| ≤ |||v||| |||u|||
(u, v)7→ v◦uL(E, F )× L(F, G)
L(E, +, ., ◦)
(A, +, ., ×)A A
a, b ∈A||a×b|| ≤ ||a|| ||b||
uL(E, F )L(E, F )v∈ L(F, E)v◦u=IE
u◦v=IFH(E, F )Isom(E)E=FL(E, F )L(E)
u∈ L(E)|||u||| <1un
IE−uL(E) (IE−u)−1=
∞
P
n=0
un
H(E, F )L(E, F )
Isom(E)L(E)