Espaces vectoriels normés II

n∈N∗
n(a0, . . . , an−1)∈CnP(X) = Xn+an−1Xn−1+
. . . +a0z∈CP
|z|6max (1,
n−1
X
i=0
|ai|).
SnR[X]R
SnRn[X]
GR
a∈ZG=aZ
GR
θ∈R\Q{e2iπnθ, n ∈Z}S1
S1
E A ⊂E A 6=∅
x∈A(xn)A x
A A
A
E={f∈ C0(R)/ f }
A={f∈E / lim+∞(f) = 0}A E

E f ∈ L(E)
∀(x, y)∈E2, < x, y >= 0 ⇒< f(x), f(y)>= 0.
k∈R+∀x∈E||f(x)|| =k||x|| k∈R+
1
kf
Φ : R[X]2→R[X]
(P, Q)7→ R∞
0e−tP(t)Q(t)dt .
Φ
inf(a,b)∈R2RR+e−t(t2−at −b)2dt
Q R
◦
Q Q
E={1/n, n ∈N∗}ER
◦
E E
GLn(C)Mn(C)
Mn(C)
E=C0([0,1],R)N1N2E
N1(f) = sup
x∈[0,1]
|f(x)|, N2(f) = Z1
0
et|f(t)|dt.
N1N2E
(fn)n∈NE
fn(x) = 1−nx 06x61/n
0
N1N2