205. Espaces complets. Exemples et applications

(E, d) (F, ||.||) (H, <, >)
(xn)E∀ε > 0,∃N∈N|n, m >N→(xn, xm)>ε
(xn)→(xn)→(xn)
Rxn= (−1)nE= (0,1] 1/n
E
(E, d)E
R C (0,1) Q
d(x, y) = |x−y|d(x, y) =
|1/x −1/y|(0,1]
a∈¯ A∀ε > 0,∃α > 0|(x, y)∈B(a, δ)∩A⇒
d(f(x), f(y)) 6ε
⇒
S1
Rn
E F E E/F

E F E
f E F
E F F
E E
G E
D(Ω) LpLp
D(Rn)HsHsD(Ω) HsHs
0
HsΩ
⇒ L(E, F )
H(Ω)
Lp(X, T, µ)
E E
t x
yt=f(t, x)y(0) = y0
x(t) = ϕ(t)+
b
R
a
K(s, t)x(s)ds
x∈C0([a, b],C) (b−a)||K||∞<1
f:X×E→E
λ7→ fλ, .)
0RnC1
f

A¯ A
C(X, Y )
L1∩L2
L2L2F
L2
G−δ
G−δ
Ti||T i|| <∞,
G−δ sup||T i(x)|| = +∞
E×F G E, F, G
G−δ
f(B)
Rg(x)6p(x)
g g
||x|| ||f(x)|| =
||x||2
||x|| =sup(|f(x)|) = max(|f(x)|)f
Jb(ϕ) = ϕ(b)|||Jb||| =||b|| Jb
b7→ JbE E00

||e|| = 1 ||uv|| 6||u||||v||
L(E, F )l1(N)
||x|| <∞Pxn
(1 −x)−1=
∞
P
0
xn||1−x|| 61/(1 − ||x||)
B(u, ||u||−1)
C
E exp(x) :=
∞
P
0
xk
k!
E E ||ex|| 6e||x|| A B
onaea+b=eaebeAe−Af C∞
A t exp(tA)t=exp(tA)
I ef(t)
t=f0(t)e(f(t))
Rny0=Ay y(0) = y0exp(tA)y0.
L2(Ω) l2(Z)Hs
ϕ∈H0ϕ= (f, .)
||x+y||2+||x−y||2= 2(||x||2+||y||2)
(f−p(f), y −p(f)) 60
en||
n
P
0
(e, ek)|| 6
||e|| ||e|| =P0∞(ek, e)||
ϕinH0u∈K a(u, v −u)>ϕ(v−u)
a/2a(v, v)−ϕ(v)
ϕinH0
u∈K a(u, v −u) = ϕ(v−u)
Ωf∈L2
−∆u+u=f
u= 0
H0
1(Ω)

R