Espaces complets - Université de Caen

R
d1(x, y) = |x3−y3|, d2(x, y) = |ex−ey|, d3(x, y) = log(1 + |x−y|).
(xn)d1(x3
n)
d3
tlog(2) ≤log(1 + t)≤t t ∈[0,1]
N∗
d1(k, l) = |k−l|, d2(k, l) = δk,l(1 + k−1+l−1), d3(k, l) = |k−1−l−1|.
f:N∗→N∗k7→ k+ 1 d2
d3X=N∗∪{∞} d3(k, ∞) = d2(∞, k) = k−1k∈N∗(X, d3)
d1d2d3
(E, d) (E, N)
E={n2, n ∈N}d(k2, l2) = |k2−l2|
E={(x, y)∈R2|x2y≤cos(x+y)}d((x, y),(x0, y0)) = |x−x0|+|y−y0|
E=Mn(C)N(A) = pTr(t AA)
E=On(R)d(A, B) = pn−Tr(t AB)
E={(xi)∈Rn|Pxi= 0}N((xi)) = Pi|xi|
E={f∈C([0,1] ,R)|R1
0f(t)2dt ≤1}d(f, g) = kf−gk∞
Q={rn, n ∈N}rn6=rmn6=m
X=R\Qn∈Nx y ∈X
dn(x, y) = 
|x−rn|−1− |y−rn|−1 , d0
n(x, y) = dn(x, y)
1 + dn(x, y)
d0(x, y) = |x−y|+
∞
X
n=0
2−nd0
n(x, y).
dnd0
nd0X
d0R
x∈X  > 0α > 0Bd(x, α)⊂Bd0(x, )
(xk)k∈XNd0
(xk)x∈R
x X (|xk−rn|−1)kn
d0(xk, xl)d0(xk, x)l→ ∞
(xk)x d0
X d0
E=`1(N,[0,1]) u= (u(k))k∈N[0,1] (Pku(k))
E
d1(u, v) =
∞
X
k=0 |u(k)−v(k)|.
E(un)nE
k∈N(un(k))nR
v:N→[0,1] limn→∞ un(k) = v(k)k
P∈Nd1(up, uq)≤ p q ≥P
PN
k=0 |up(k)−v(k)| ≤  N ∈N
v∈E d1(up, v)≤ p ≥P

E=C1([−1,1] ,R)E
N(f) = kfk∞, P (f) = kfk∞+kf0k∞,
k·k∞
fn(t) = √n−1+t2n∈Nt∈[−1,1]
g(t) = lim fn(t)t∈[−1,1]
(fn)ng
E N
(fn)n∈ENC1fn(0) (f0
n)n
g(fn)nf f0=g
E P
f: [0,1] →RC
|f(x)−f(y)| ≤ C|x−y|x y ∈[0,1] Lip(f)C
Lip(f) = sup
x6=y
|f(x)−f(y)|
|x−y|.
E[0,1] R0
Lip E
f∈Ekfk∞≤Lip(f)
(fn)n(E, Lip)
(fn)nf: [0,1] →R
(fn)nE f : [0,1] →R
Lip(fp−fq)≤L p q Lip(fp−f)≤L f ∈E
(E, Lip)
E=R[X]kPk=Pn
i=0 |ai|P=Pn i=0 aiXi
k ϕk:P=P∞
i=0 aiXi7→ ak
Sn
(Sn)nE E
(Pn)ndeg(Pn) +∞
d(Pnk)kdeg(Pnk)≤d k
(Pn)nE
deg(Pn) +∞
(x, y)∈R2
x= (sin x+ 8 cos y)/10
y= (2 cos x+ 7 sin y)/10.
f:R2→R2
k·k∞k·k1
(X, d)f:X→X
n∈N∗fn=f◦f◦ ··· ◦ f
x fnf(x)fn
f
¯
B=¯ B(0,1) R2N∞: (x, y)7→ max(|x|,|y|)
6x= 1 −x2+ 3y,
6y=x+ 2y2.
¯
B¯ B N∞
(x, y) = f(x, y)
f¯ B
¯
B
R2
fR2