202: exemples de parties denses et applications

X¯ Y=X
x X Y x
F([0,1],R)
(X, d)
K
S1
E
d(det(X))(H) = T r(t(ComA)H))
ln(2) = P∞
1(−1)(n+ 1)/n
R C
χAB =χBA

R
Q R
ERf
f
RAZ
p q p +qZ
R
sin(n) [−1,1]
cos(n)
Lp(Rn)p=∞
E ei¯ vect(ei) = E
G−δ
G−δ
E T
Tn(x)E E
X Y S Y Y
X T n(x) 0
Y
Y T S =Id
HC(T)E
ϕ C∞
ϕ(n)(x)=0 ϕ
fR+n f(nx)
∞

Ti
||T i|| <∞, G −δ sup||T i(x)|| = +∞
E×F G E, F, G
G−δ
E F F X
E X f E
K Y
E F X E
X F |||f|||
f E |||f|||
L1(Rn)∩L2(Rn)L2(Rn)
L2
K A C0(K, R)
X∀x f ∈A f(x)6= 0 A C0(K, R)
aifi||x−ai||
R2
C∗
E
E E Rn
C(K, K)
Ck
[a, b]

fRRftn=
0f f RRft2n= 0
f f RRft2n= 0
f= 0 fR+In:= Rf(t)e−nt = 0
n f
f u
Rfu = 0 f f
Rfu = 0 f= 0
P(x)e−x
R+0 +∞
un[0,1] 0 6a6b61
Xn(a, b) = Card {k6n|uk∈[a, b]}
Xn(a, b)/n
f f(uk) 0 1 f
lim
n→∞
1
n
n
X
1
e2ipukπ= 0
L2[0; 2π]
A L(H)
A A
L1
f+f−
f L1RRnf(x)eitxdx →0
∞ −∞
X(Ω,A, µ)h
R R h(X)∈L1(Ω) ⇔h∈L1(R)RΩh(X(Ω))dµ =
Rh(x)dPX(x)dPXX dPX(B) = µ(X−1(B)B
(Ω, A, P )X Y
X Y L1XY L1
E(XY ) = E(X)E(Y)X Y L2Cov(X, Y ) = 0
V ar(X+Y) = V ar(X) + V ar(Y)

C∞Lp(Rn)p < ∞
C∞Ω Ω Rn
f∈Lp(Rn)
16p < ∞lim
y→0RRn|f(x+y)−f(x)|dx = 0
f g L2(Rn)f∗g x
||f∗g||∞6||f||2||g||2
f∈L1
loc(Rn)
C∞Rf(x)u(x)dx = 0 f= 0
L1
loc
L(1)
Rf W 1,p
Lpf u Ry
xf0(x) =
u(y)−u(x)