feuille de TD n° 6 - Université de Caen

◦
(X, d)f:X→X
n∈N∗f◦nf n
f
RE:= C0([0,1],R) [0,1] R
k·k∞
f:E−→ E
x7−→ t7→ 1
21 + Z1
0
testx(s)ds
f x kx−xnk∞xn
E x0:t7→ 1xn+1 =f(xn)n∈N
C0
b(R,R)R R R λ∈]0,1[
g∈ C0
b(R,R)
f∈ C0
b(R,R)
∀x∈R, f(x) = λf(x+ 1) + g(x).
f g
(E, k·k)K E F E
F(k, f)∈K×F d(k, f ) = d(K, F )
(k1, k2)∈K2d(k1, k2) = (K)
E=Rn(k, f)∈K×F d(k, f ) = d(K, F )

(E, d) (Kn)n∈N
E n Kn+1 ⊆Kn
∩n∈NKn
(un)n∈NE n ∈Nun∈Kn
(uϕ(n))n∈N(un)n∈Nn∈Nuϕ(n)∈
Kn
∩n∈NKnE
O E ∩n∈NKnn∈Nkn⊆O
f g X f
X g−1(] − ∞,0]) A > 0
∀x∈X, A ·f(x) + g(x)>0.
An={x∈X|n·f(x) + g(x)≤0}
(E, k·k)A B E
A+B:= {a+b|(a, b)∈A×B}.
A A +B
A B A +B
A B A +B
R×R
A:= {(a, b)∈R×R|a > 0ab = 1}B:= {(a, b)∈R×R|a= 0 b≤0}.
A, B A +B
n∈N∗RMn(R)n×n
k · k∞A(ai,j )1≤i,j≤nkAk∞= max1≤i,j≤n|ai,j | Mn(R)
On(R)
R2
A:= {(x, y)∈R2|x2−y2−2xy ≤1}
B:= {(x, y)∈R2|x2+y2+exy ≤36}
C:= {(x, y)∈R2|2x2+ 3y2<1}
(E, d) (xn)n≥0E a ∈E
K={xn|n≥0}∪{a}E
(E, d) (xn)n≥0E
a E a