T
M
M
T (x) = x x
T T
M
X
∅, X
τ ⊆ P (X)
X
∈ τ,
{U α }α ⊆ τ
U α ∈ τ,
α
n
{U k }nk=1 ⊆
τ
U k ∈ τ
k=1
τ
∅
X
(X, τ )
τ A⊆X
X
x ∈X
x ∈ U ⊆ A
U ∈ τ
ET
S X
X
S
X
τ
S (X, τ ) x1 , x2
U 1
X
(X, τ )
U 2
x1
x2
X
(X, τ ) C
x x
X
X
X
Y
X
Y
U X\Y
x0 U y
V y
Y
x0
n
V = x0
Y
V yi
n
U =
i=1
Y
X
x∈X
{V y }y∈Y
y Y
1
Y ⊆ V. y ∈ Y
X
X
V y ,...,V yn
V
x0 x0 ∈ U
U yi
U
i=1
X
U
x
V
x
V
V
V ⊂ U
⇐)
C = V
⇒)
C
V x∈X C = Y\U
X Y
X
C V
Y
V ∩ C =
U
x C
Y
x ∈ V
W V ⊂ U
∅
C ⊂ W
V
Y
(X, τ ) F n◦ = ∅
n∈N
◦
F n
n
◦
F n
n
x
X
{F n }n∈N = ∅.
∈N
∅ F n◦ =
=∅
X
n ∈ N.
∈N
{Gn }n∈N
n
Gn
∈N
F n Gn = X\F n
n∈N
Gn = X\F n◦
n
x∈X
U
U ∩ G1 V 1
Gn = X\
∈N
x
n
F n = X\
∈N
n
.
∈N
X U ∩ G1 =∅
G1
X
V 2
F n
◦
V 1 ⊂ V 1 ⊂ U ∩ G1 V 1 ∩ G2
G2 V 2
y ∈ U ∩ G1 V 1 X V 1 ∩ G2 =∅ V 2 ⊂ V 2 ⊂ V 1 ∩ G2
y
V 2 ⊂ G1 ∩ G2
{V n }n∈N n∈N
V n
V n+1 ⊂ V n+1 ⊂ V n ⊂ V n ⊂ V 1 ⊆ U V n ⊆
k
z∈ x∈
n
∈N
∈N V n
n
Gn
Gk
∈N
V 1
V n
z ∈ U ∩
Gn
n
n
∈N
U
x
∈N
(X, τ )
X
τ
F
X×X →X F×X →X
(x, y) → x + y (α, x) → αx
EV T
αA ⊆ A
A
α∈F x∈X
A
|α| ≤ 1; tx ∈ A
ε > 0
[a, b] = {(1 − t)a + tb : 0 ≤ t ≤ 1} ⊆ A A⊆X t>s
X
A ⊆ t U
0 < t < ε.
A ⊆ X a, b ∈ A
F
X
s > 0
A⊆X
F
X
A
A
U
V
X
V
{rn }n ⊆
rn −−−→ ∞
F
n
V (α, x) → αx α∈F |α| < δ
rn
rn V.
n=1
×X→X αW ⊆ V
F
δ > 0 U =
W αW
U
|α|<δ
V
1
X=
→∞
∞
x ∈ X A = {α ∈ F : αx ∈ V } ∈ A, ∀n ≥ n0
h :
F
1
rn x
→
F × X
h(α) = α x
∈ V
x∈X
x ∈ rn V X=
∞
n0 ∈ N n ≥ n0
rn V.
n=1
EV T X
EV T K ⊆ X
V W
V
K ⊆
∞
nW
n=1
K W t > nm
n1 < n2 < ... < n m n j W ⊆ n i W
j
K ⊆ n m W
K ⊆
m
ni W
i=1
K ⊆ tW ⊆ tV
EV
X
p : X → [0, ∞)
F
p (x + y) ≤ p (x) + p (y), p (αx) = |α| p (x). x =0
p (x) = 0
P
X
U
n
τ X p ∈ P x0 ∈ X ε > 0 p1 ,...,pn ∈ P ε1 ,...,ε n > 0 X
{x : p (x − x0 ) < ε} x0 ∈ U
X
{x ∈ X : p i (x − x0 ) < εi } ⊆ U
EV T
X
i=1
P
X
{x : p (x) = 0} = {0}
∈P
p
p
P p (x − y) = 0 V = z : p (y − z) < 2ε
{x : p (x) = 0} = {0}
∈P
x, y ∈ X ε > 0
p (x − y) > ε U V
U = y
x
K
p ∈ P z : p (x − z) < 2ε
X
A
x, y ∈ K
K
[x, y] ∩ A = ∅ ⇒ x, y ∈ A [x, y] = {(1 − t)x + t y : t ∈ (0, 1)} z ∈ K
{z } x, y ∈ K z ∈ [x, y] ⇒ x = y = z. Ext (K )
K F
X
X = {ϕ : X → F : ϕ (X, τ )
X
} X X∗ = (X, τ )∗ = τ
X∗τ
(α ϕ + ψ)(x) = α ϕ(x) + ψ(x) C
X R
x∈X
ϕ ∈ X : ϕ α∈F
τ ϕ, ψ ∈ X∗τ
XR
.
(X, τ )∗R
(α ϕ + ψ) ∈ X∗τ
X∗τ
ELC
X X
{ pϕ : ϕ ∈ X∗τ } ,
pϕ (x) = |ϕ(x)| , x ∈ X ”wk”
X
σ(X, X∗τ )
X∗τ
px (ϕ) = |ϕ(x)| , ϕ ∈ X∗τ
{ px : x ∈ X} , X∗τ
”wk ∗ ”
U ϕ1 ,...,ϕn ∈ X∗τ
σ(X∗τ , X) x0 ∈ U
X
ε > 0 n
{x ∈ X : |ϕi (x − x0 )| < ε} ⊆ U.
i=1
{xi }i∈I ⊆ X
x0
ϕ(xi ) X∗τ
U
x1 ,...,xn ∈ X
ϕ ∈ X∗τ
ϕ(x0 )
ϕ0 ∈ U
ε > 0 n
{ϕ ∈ X∗τ : |(ϕ − ϕ0 )(xi )| < ε} ⊆ U.
i=1
{ϕi }i∈I ⊆ X∗τ
ϕ
ϕi (x)
x∈X
ϕ(x)
S 1 ⊂ K
S 2 ⊂ S 1
K
S 2
S 1
K x, y ∈ K t ∈ (0, 1) K x, y ∈ S 1
(1 − t)x + ty ∈ S 2 S 2
S 2 ⊂ S 1 S 1
S 1
x, y ∈ S 2
A⊂X
X
∅ Af = {x ∈ A : f (x) = s } =
f ∈ X
{f (x) : x ∈ A} < ∞
s =
Af
A
x, y ∈ A t ∈ (0, 1) (1 − t)x + ty = z ∈ Af f (z) = (1 − t)f (x) + tf (y) < s y∈ / A
f (x) < s
x∈ / Af f (z) < s
z∈ / Af
A
X
co(A) = X
co(A) = co(A)
A
X
{C : A ⊆ C,
{C : A ⊆ C,
co(A).
}. A
}.
(X, )
{x1 ,...,xn } ⊆ X
n
co({x1 ,...,xn }) =
n
αi xi :
i=1
αi ∈ [0, 1]
αi = 1
i = 1,...,n
i=1
n
(α1 ,...,α n ) ∈ Rn :
αi ∈ [0, 1]
αi = 1
i = 1,...,n
i=1
n
(α1 ,...,α n ) →
co({x1 ,...,x n })
αi xi
i=1
(X, τ )
EV T
U
X
U V ϕ ∈ (X, τ )∗
t∈R x ∈ U, y ∈ V.
Re(ϕ(x)) < t < Re(ϕ(y)) (X, τ ) K
X
ELC
K
F
ϕ ∈ (X, τ )∗R ε > 0
F
ϕ(x) ≤ t − ε < t ≤ ϕ(y)
t∈
R
x ∈ K, y ∈ F. ELC K K = co(Ext(K )). X
X
Ext(K ) =∅
Ext(K ) =∅ K B ⊆ A C C
A≤B
A
C A, B ∈ C K ∈ C S =
A
A
∈A
A
S =
K K K
(1 − t) x + t y ∈ A x, y ∈ A
B ≤ M (M ⊆ B) M M f
∅
K A ∈ A A∈A C B ∈ C f ∈ X∗τ K
S x, y ∈ K
t ∈ (0, 1) A x, y ∈ S
A
(1 − t) x + t y ∈ S
A M ∈ C
M f f ∈ X∗τ
M M
M f = M, ∀f ∈ X∗τ M
f ∈ X∗τ
K
K = co(Ext(K ))
⊇)
X∗τ
M
H = co(Ext(K ))
H ⊆ K
K
⊆)
x0 ∈ K \H
H
{x0 } t ∈ R ε > 0
f : X → R
H x ∈ H
f (x) ≤ t − ε < t ≤ f (x0 ). f (H ) < t ≤ f (x0 ) K e
K f = {x ∈ K : f (x) = x∈K f (x)} ⊆ K K f K K f ∩ Ext(K ) =∅
e
f (e) =
x K f (x)
∈
≥ f (x0 ) ≥ t > f (Ext(K )). K ⊆ co(Ext(K ))
K 1 ,...,K n co(K 1 ∪ ... ∪ K n ) = co(K 1 ∪ ... ∪ K n )
X X
co(K 1 ∪ ... ∪ K n ) co(K 1 ∪ ... ∪ K n )
n
K = x, y ∈ K
X
n
αi xi : α i ∈ [0, 1],
i=1
α ∈ [0, 1]
xi ∈ K i , i = 1,...,n .
αi = 1
i=1
n
n
αx + (1 − α)y = α
αi xi + (1 − α)
i=1 n
β i yi
i=1
(ααi xi + (1 − α)β i yi )
=
i=1
n
αi , β i ∈ [0, 1]
n
αi = 1,
i=1
ri = αα i + (1 − α)β i
xi , yi ∈ K i
β i = 1
i=1
i = 1,...,n
n
i = 1,...,n
ri = 1
i=1
n
αx + (1 − α)y =
i=1
K
ααi xi + (1 − α)β i yi ri , ri
−α)βi yi ∈ K i K i ααi xi +(1 ri K = co(K 1 ∪ ... ∪ K n )
αx + (1 − α)y ∈ K
{ei }ni=1
R
n
n
C =
n
αi ei : α i ∈ [0, 1]
i=1
αi = 1 .
i=1
f : C × K 1 × ... × K n → K n
f ((α1 ,...,α n , x1 ,...,xn )) =
αi xi .
i=1
C × K 1 × ... × K n f (C × K 1 × ... × K n ) = K
f
F ⊂ K
K
X X
K
Ext (K ) ⊆ F
K = co(F )
x0 ∈ Ext(K )\F
F F F ∩ {x ∈ X : p (x − x0 ) < 1 } = ∅ x0 ∈ / (F + U 0 )
x0 U 0 = x ∈ X : p (x) <
y1 ,...,y n ∈ F
F
1 3
X p (x0 + U 0 ) ∩ (F + U 0 ) = ∅
n
F ⊆
(yk + U 0 )
i=1
K k = co(F ∩ (yk + U 0 ))
K k ⊆ yk + U 0 K k ⊆ K K 1 ,...,K n co(K 1 ∪ ... ∪ K n ) = co(K 1 ∪ ... ∪ K n )
K = co(F ) = co(K 1 ∪ ... ∪ K n ) x0 ∈ K
n
x0 =
n
xk ∈ K k , αk ≥ 0
αk xk
k=1
K
x0 = xk
αk = 1
x0
k=1
k
x0 ∈ K k ⊆ yk + U 0 ⊆ F + U 0
d
1≤d<∞ B x
f : B → B
R
B
f (x) = x
K B ⊂
n
f : K → K h := g ◦ f ◦ g −1 : B → B h(x) = x y = g −1 (x)
R
x∈B
g : K → B f
H H
K k0 ∈ K
h∈H
h − k0 =
(h, K ) ≡
h = 0 ˜ := K − ˜h = K
˜ ∈ H h
k˜
k0 = ˜k + ˜h h = 0 d := {kn }n ⊆ K
n∈N
kn − km 2
kn +km K 2 2 1 2 2 ∀n ≥ n0 kn < d + 4 ε
{kn }n kn − k0 −−−→ 0 n
→∞
{h − k : k ∈ K } .
(0, K ) =
k − ˜h : k ∈ K
k˜ =
2
h = 0
˜) = (0, K
{k : k ∈ K } d
1 kn + km = kn 2 + km 2 − 2 2
−
kn km
2
kn − km 2
kn +km
∈ K
2
<
1 2
2
2
≥ d2
2
ε > 0 n, m ≥ n0
n0 ∈
N
1 1 2d2 + ε2 − d2 = ε2 . 2 4
kn −−−→ k0 n
˜ k : k ∈ K
→∞
H
K
k0 ∈ K k0 = d
k˜0 ∈ K ˜0 k0 +k 2
k0 = k˜0 = d
∈ K
k0 + ˜ k0 1 d≤ ≤ k0 + k˜0 2 2
2
k0 + ˜ k0 d2 = 2
˜0 k0 k
− 2
2
k0 = ˜k0 .
=0
= d,
k0 − k˜0 = d 2 − 2
2
.
K X f : K → K x
K
f (x) = x Cd
X
R2
d
K = x ∈ R : x ≤ r g : K → B g(x) = xr K φ : B → K x − yK (x, K ) φ(x) = x x ∈ K φ ε > 0
n0 ∈
N
d
R
d
X = Rd
R
d
φ(x) = y K φ
d(xn , φ(xn )) ≤ d(xn , φ(x))
yK
B
1≤d<∞
K ⊆ B ≡ x ∈ Rd : x ≤ r K
r > 0
{xn }n d(xn , x) < 2ε , ∀n ≥ n 0
x ∈ B n ≥ n 0
xn −−−→ x n
→∞
d(x, φ(x)) ≤ d(x, φ(xn )).
ε d(xn , φ(xn )) ≤ d(xn , φ(x)) ≤ d(xn , x) + d(x, φ(x)) ≤ + d(x, φ(x)). 2
d(x, φ(x)) ≤ d(x, φ(xn )) ≤ d(x, xn ) + d(xn , φ(xn )) ε < + d(xn , φ(xn )) < ε + d(x, φ(x)). 2
d(x, φ(x)) ≤ d(x, φ(xn )) < ε + d(x, φ(x)). n
→∞ d(x, φ(xn )) = d(x, φ(x)).
Rd
φ
{φ(xn )}n ⊆ K y ∈ K φ(xnk ) → y d(x, φ(xnk )) d(x, y) φ y = φ(x) φ φ(x) = x x∈B x ∈ K x = f (φ(x)) = f (x)
{φ(xnk )}k
K d(x, y) = d(x, φ(x)) φ f ◦ φ : B → K ⊆ B (f ◦ φ)(x) = x
(f ◦ φ)(B) ⊆ K
H := L 2 (−π, π) en (t) =
T : H → H T
xn en =
n
∈Z
n
xn en+1 ,
∈Z
f : H → H
f (x) =
x ≤ 1 f (x) ≤
n
∈Z
2
0 + T (x)
f
x ˜ ∈ B 1 (0) f (˜ x) = x ˜ f (˜ x) = T (˜ x) T (x) ˜ =x ˜ T x = 0 x ˜ ∈ H xn en xn = x, en x ˜ = T (˜ x)
−x e
1
1 − x 1 − x + T (x) ≤ + x ≤ 1, 2 2
f (B1 (0)) ⊂ B1 (0)
x ˜ =
{en }n∈Z
eint
√ 2π
x ˜ =0
1 − x ˜ 1 − x˜ x ˜ − T (˜ x) = x ˜ − f (˜ x) − e0 = e0 , 2 2
(xn − xn−1 ) en =
∈Z
n
xn − xn−1 =
y ∈ K
0
−x˜
1
2
1 − x ˜ e0 . 2
n =0 . n = 0
x˜ < 1
n
x0 = xi , ∀ j ≤ −2 x j = x−1 = x2n = x ˜2 < ∞.
∈Z
H = l2
B =
∀i ≥ 1
x ∈ l2 : x ≤ 1
f (x) = ( 1 − x2 , x1 , x2 ,...) f (˜ x) = x ˜ x) = x ˜ = 1 f (˜
f
f : B → B
f (x) = 1
x ˜∈B
f
f (˜ x) = ( 1 − x ˜2 , ˜ x1 , ˜ x2 ,...) = (0, ˜ x1 , ˜ x2 ,...) = x ˜. 0=x ˜1 = x ˜2 = ...
x ˜ = 0
f E ⊆ X
X
f
f (A)
A E A f (E )
K
f : E → X E
K K ⊆
f : E → X
X
E
f (A)
X ε > 0
{B(a, ε) : a ∈ A} φA (x) =
ma (x) = 0 x − a ≥ ε φA (x) − x < ε
A
φA : K → X
{ma (x)a : a ∈ A} , {ma (x) : a ∈ A}
ma (x) = ε −x − a x ∈ K.
x − a ≤ ε
φA
ma (x) ≥ 0
{ma (x)a : a ∈ A} > 0 a ∈ A K φA a∈A ε˜ > 0 x1 ∈ B(a, ε) x2 ∈ K \B(a, ε) d(x1 − x2 ) < δ
x ∈ K φA ma : K → [0, ε] 0 < δ ≤ ε˜
|ma (x1 ) − ma (x2 )| = |ε − x1 − a| ≤ |x2 − a − x1 − a | ≤ x2 − x1 < δ ≤ ˜ε.
φA (x) − x =
x − a < ε
ma (x) > 0
φA (x) − x ≤
{ma (x)(a − x) : a ∈ A} , {ma (x) : a ∈ A}
{ma (x) a − x : a ∈ A} < ε. {ma (x) : a ∈ A}
E X
x ∈ E
f (x) = x
f : E → X
f (E ) ⊆ E
K = f (E ) K
An φn (K ) ⊆ co(K ) ⊆ E
K ⊆
f n ≡ φn ◦ f
K ⊆ E n ∈ {B(a; 1/n) : a ∈ A} φn = φAn φn E E
N
f n (x) − f (x) < 1/n x ∈ E E n = E ∩ Xn
An
Xn
Xn
E n E n
Xn
xn ∈ E n
f n (xn ) = x n
f (xnj )
f n : E n → E n
Xn
j
{f (xn )}n∈N
∈N
x0 ∈ K
K f nj (xnj ) = xnj
f (xnj ) −−−→ x0 j
→∞
xnj − x0 ≤ f nj (xnj ) − f (xnj ) + f (xnj ) − x0 ≤
xnj −−−→ x0 j
→∞
E
H0 = x ∈ H
1 2 {xn }∞ n=1 ∈ : |xn | ≤ 2
n−1
ϕ(x) = x
X
A B0 (X, Y)
f (x0 ) =
→∞ f (xn
n
A : X → Y B (X)
Y Y
X
X Y B0 (X) = B0 (X, X)
T ∈ B(X) M X T (x) ∈ M x ∈ M (0) X Lat(T ) A ⊆ B(X) Lat(A) = {Lat(T ) : T ∈ A} X
co(K )
) = x 0
ϕ : H → H
X
X
j
H
A(B (X))
T
1 + f (xnj ) − x0 . n j
K
T M
co(K ) ε > 0 n ε K ⊆ B(x j , ) 4
K
x1 , x2 ,...,xn
K
C = co {x1 , x2 ,...,xn }
C
j =1
m
y1 , y2 ,..,ym ∈ C
C ⊆
ε B(yi , ) 4
w ∈ co(K )
i=1 l
z ∈ co(K )
w − z <
ε
z =
4
x j (r) − zr <
x j (r) l
z−
r =1
ε
4
l
αr (zr − x j (r) ) ≤
αr x j (r) =
r=1
r=1
l
l
αr x j (r) ∈ C
yi ∈ C
r =1
zr ∈ K, αr ≥ 0
αr zr
zr
r =1
co(K ) ⊆
B(yi , ε)
αr = 1
r =1
l
αr zr − x j (r) <
r=1
αr x j (r) − yi <
m
l
ε . 4
ε
4
co(K )
i=1
B(X)
A
I ∈ A Lat(A) = { (0), X} ker(AK − I ) = (0).
K
A ∈ A
X
K K K = 1 K (x0 ) > 1 x0 ∈ X K S = { x ∈ X : x − x0 ≤ 1} x0 0∈ / S 0 ∈ S 1 < K (x0 ) ≤ K x0 ≤ 1 x0 ≤ 1 K (x0 ) − K (x) ≤ K x0 − x ≤ 1 K (x0 ) > 1 x ∈ S K (S ) 0∈ / K (S ) K K (S ) A x ∈ X {T (x) : T ∈ A} A I ∈ A x = I (x) ∈ {T (x) : T ∈ A} {T (x) : T ∈ A} ∈ Lat(A) {T (x) : T ∈ A} = X T (y) − x0 < 1 x ∈ X y ∈ K (S ) T ∈ A {y : T (y) − x0 < 1 } {y : T (y) − x0 < 1 } K (S ) ⊆
∈
T A n
K (S ) ⊆
T 1 , T 2 ,...,T n ∈ A
K (S )
{y : T j (y) − x0 < 1 }
y ∈ K (S )
1 ≤ j ≤ n
j =1
a j (y) = T j
a j (B1 (x0 ))c
{0, 1 − T j (y) − x0 } . B1 (x0 ) a j
n
y ∈ K (S )
a j > 0
a j (y) > 0
j =1
b j : K (S ) → R a j (y)
b j (y) =
n
a j (y)
j =1
a j
ψ : S → X
n
ψ(x) =
b j (K (x))T j K (x).
j =1
b j , K T j ψ(S ) ⊆ S x ∈ S K (x) ∈ K (S )
T j (K (x)) − x0 < 1
ψ
b j (K (x)) > 0
T j K (x) ∈ S
a j (K (x)) > 0
n
b j (K (x)) > 0
b j (K (x)) = 1
j =1
x ∈ S
ψ(x) ψ(S ) ⊆ S T j K ∈ B0 K K (B ) T j (K (B)
S j
S
B T j T j (K (B))
X
T j (K (B)) n
T j K (S )
j =1 n
co(
T j K (S ))
ψ (S )
j =1
S
T j K (S )
ψ (S ) ψ x1 ∈ S
n
β j = b j (K (x1 ))
A =
ψ (x1 ) = x 1
A∈A
β j T j
j =1
n
A(K (x1 )) =
β j T j (K (x1 )) = ψ(x1 ) = x 1 .
j =1
x1 =0
0∈ / S
x1 ∈ ker(AK − I )
ker (AK − I ) =0
T ∈ B(X) AM ⊆ M
A
T
T
T
T K = K T
M X AT = T A
T T ∈ B(X) K
AM ⊆ M
T T
A = T Lat(A) = { (0), X}
T I
A
A∈A
N ∈ Lat(AK )
N = ker(AK − I ) = (0) x ∈ N
AK | N
AK (x) = AK (x) − x + x = (AK − I )(x) + x = x. B N (0, 1) = AK (B N (0, 1))
N AK (T (x)) = T (AK (x)) = T (x) N T | N T = λI x∈M
AK ∈ B0 AK | N ∈ B0 N B N (0, 1) A K T x ∈ N T (x) ∈ N T N M = ker(T − λI ) M = X λ = (0) T S ∈ A
(T − λI )(S (x)) = T S (x) − λS (x) = S (T (x)) − λS (x) = S ((T − λI )(x)) = S (0) = 0. S (x) ∈ M T
S (M) ⊆ M
M
K
n
T (
α j x j ) =
j =1
V
n
x j ∈ K α j ≥ 0
α j T (x j )
T : K → V
n
j =1
α j = 1
j =1
K
F
X
T : K → K T ∈ F
x0 ∈ K
T ∈ F
T (x0 ) = x 0 T (n) : K → K
n≥1 (n)
T
1 = n
−
n 1
T k .
k=0
n, m ≥ 1 S (n) T (m) = T (m) S (n) S (n−1) T = T S (n−1) n∈N
S, T ∈ F ST = T S
F
S (n) T = SS (n−1) T = ST S (n−1) = T SS (n−1) = T S (n) . S (n) T (m−1) = T (m−1) S (n)
m∈N
S (n) T (m) = S (n) T (m−1) T = T (m−1) S (n) T = T (m−1) T S (n) = T (m) S (n) .
K = T (n) (K ) : T ∈ F , n ≥ 1 T T K K n) ( T (K ) T 1 , T 2 ,...,T r ∈ F n1 , n2 ,....,nr ≥ 1 r
F
(n ) (n ) T 1 1 ...T r r (K )
⊆
j =1
T ∈ F
K x0 ∈ T (n) (K ) (n)
x0 = T
(nj )
T j
(K )
K
K
x0 ∈ {B : B ∈ K} n≥1
x ∈ K
1 (x) = x + T (x) + .... + T (n−1) (x) . n
1 1 T (x0 ) − x0 = T (x) + ... + T (n) (x) − x + T (x) + ... + T n−1 (x) n n 1 1 = T (n) (x) − x ∈ [K − K ] . n n
K − K
K φ(x, y) = x − y
K × K K − K
K − K 1
n ≥ 1
U T (x0 ) − x0 ∈ U
n [K −
K ] ⊆ U T (x0 ) − x0 = 0
X
F
T
F
x0
A⊆X { p (x − y) : x, y ∈ A} .
p p −
p−
X
(A) ≡
A
ELC K
X X
p
ε > 0
X
C
K
C = K p − (K \C ) ≤ ε
S = x ∈ X : p(x) ≤ A ⊆ K
ε
4
ω
D = Ext(K ) ⊂ K K ⊆ (S + a)
S
∈
D
K (an + S ) ∩ D
K
a A
a + S X
D D
D =
∞
((an + S ) ∩ D)
n=1
a ∈ A ((a + S ) ∩ D)◦ = ∅ X ∅ W W ∩ D ⊆ (a + S ) ∩ D W ∩ D = K 1 = co(D\ W ) K 2 = co(D ∩ W ) K 1 K 2 K 1 ∪ K 2 K K = co(K 1 ∪ K 2 ). K K 1 = K 1 = K K = co(D\W ) Ext(K ) ⊆ D\W D ⊆ D\W D ∩ W = ∅ D ∩ W (a + S ) ∩ D S p − { p (x − y) : x, y ∈ K 2 } ≤ 2ε K 2 ⊆ a + S (K 2 ) ≡ r ∈ (0, 1] f r : K 1 × K 2 × [r, 1] → K f r (x1 , x2 , t) = tx 1 +(1 − t)x2 f r (K 1 × K 2 × [r, 1]) C r ≡ f r (K 1 × K 2 × [r, 1]) K C r = r ∈ (0, 1] C r = K z ∈ E xt(K ) x1 ∈ K 1 , x2 ∈ K 2 , t ∈ [r, 1] z = tx 1 + (1 − t) x2 t = 0 z = x 1 Ext(K ) ⊆ K K 1 = K y ∈ K \C r y = tx 1 + (1 − t) x2
C r t ∈ [0, r)
K = co(K 1 ∪ K 2 )
x1 ∈ K 1 , x2 ∈ K 2
p (y − x2 ) = p (t(x1 − x2 )) = t p (x1 − x2 ) ≤ r d, d = p −
(K )
y˜ = ˜t x˜1 + (1 − t˜) x˜2 ∈ K \C r
p (y − ˜ y ) ≤ p (y − x2 ) + p (x2 − ˜ x2 ) + p (˜ x2 − ˜ y) ≤ 2rd + p − r =
ε
4d
C = C r
ε (K 2 ) ≤ 2rd + . 2
Q Q
X
S
X
S
Q x, y ∈ Q
0∈ / {T (x) − T (y) : T ∈ S}. ELC Q ⊆ X
X
S
Q x, y ∈ Q
S
Q
p
{ p (T (x) − T (y)) : T ∈ S} > 0.
⇒)
x, y ∈ Q
F
S {z ∈ X : p (z) < ε} ∩ {T (x) − T (y) : T ∈ S} = ∅ T ∈ S { p (T (x) − T (y)) : T ∈ S} > 0 p
p ∈ F ε > 0 p (T (x) − T (y)) ≥ ε 0 ∈ {x ∈ X : p (x) < 1 }◦
X
⇐)
x
y
Q
p
{ p (T (x) − T (y)) : T ∈ S} > 0 p (T (x) − T (y)) ≥ ε
ε > 0 U = {z ∈ X : p (z) < ε} 0∈ / {T (x) − T (y) : T ∈ S}.
T ∈ S U ∩ {T (x) − T (y) : T ∈ S} =
∅
X
Q
X
Q T (x0 ) = x 0
T
S x0 ∈ Q
Q
S S
Q Q {T 1 , T 2 ,...,T n } ⊆ S x0 ∈ Q
T 0 : Q → Q
T 0 = (T 1 + ... + T n )/n
T 0 (x0 ) = x 0 T k (x0 ) = x 0
1 ≤ k ≤ n
T k (x0 ) = x0
T k m ∈
N
x 0 T k (x0 ) =
k ≤ m
T k (x0 ) = x 0
k>m 1 1 x0 = T 0 (x0 ) = (T 1 (x0 ) + ... + T n (x0 )) = (T 1 (x0 ) + ... + T m (x0 )) + n n
n−m n
x0 .
T0 = (T 1 + ... + T m )/m 1 (T 1 (x0 ) + ... + T m (x0 )) m n 1 = (T 1 (x0 ) + ... + T m (x0 )) mn n n−m = x0 − x0 m n = x 0 .
T 0 (x0 ) =
T k (x0 ) = x 0
1 ≤ k ≤ n
T 0 (x0 ) = x 0
S
p T ∈ S
S 1
{T 1 ,...,T n }
C S (x0 ) ∈ K \C
1
K
S (x0 ) = ST 0 (x0 ) =
(∗)
S 1 ⊆ S S 1 = {T l ...T lm : m ≥ 1, 1 ≤ l j ≤ n} S K = co {T (x0 ) : T ∈ S 1 } K Q C = (K \C ) ≤ C = K p − K T 0 (x0 ) = x 0
S 1
1 [ST 1 (x0 ) + ... + ST n (x0 )] ∈ K \C. n 1≤i≤n
C
p (ST i (x0 ) − S (x0 )) ≤ p −
ST i (x0 ) ∈ K \C
(K \C ) ≤ ε. 1 ≤ k ≤ n
T k (x0 ) = x 0 F
QF = {x ∈ Q : T (x) = x, ∀T ∈ F } T ∈ S {QF : F ∈ F } Q {QF ,...,QF r } F i F
QF =
∈
F F
∅
S F ∈ F F ∈ F
QF F = {T : T ∈ F i QF = ∅
1
T (x) = x
ε > 0
1 ≤ k ≤ n p (T (T k (x0 )) − T (x0 )) > ε.
S ∈ S 1
X
r
T ∈ F
QF i =
i = 1, ...r} x ∈Q Q
∅
i=1
QF = ∅
x0
T (x0 ) = x 0
T ∈ S
· : G × G → G
G G
τ
G
(G, ·) (G, τ ) φ : G×G → G
φ(x, y) = x · y ψ(x) = x−1
ψ : G → G
G m
G
m(G) = 1, U
G
m(U ) > 0,
G x∈G m(A) = m(Ax) = m(xA) = m(A−1 ) Ax = {ax : a ∈ A} , xA = {xa : a ∈ A} , A−1 = a−1 : a ∈ A .
A
m
G M (G)
G G
G Q := {µ ∈ M (G) : µ(G) = 1} Rx (µ) = µ(x ·)
F = {Rx : x ∈ G}∪{Lx : x ∈ G} Lx (µ) = µ(· x)
Rx , Lx : Q → Q Q
Q
Q µ ∈ M (G)
Q
Q G
C (G)∗
F µ : C 0 (G) → F F µ (f ) =
fdµ.
G
F µ ∈ C 0 (G)∗
Φ : M (G) → C 0 (G)∗
C 0 (G)∗ = C (G)∗
G
F µ µ →
Q := Φ(Q) ⊂ C 0 (X )∗
1 Q = Φ(Q) = {F ∈ C (G)∗ : F > 0 y F (1) = 1} = U 1 (0) ∩ i− 1 (1) ∩
U 1 (0) f ∈ C (G)∗
1 + i− f (R0 ),
≥
f 0
C (G)∗ if
ω∗
if : C (G)∗ → C f ∈ C (G) U 1 (0)
if (F ) = F (f ) ω∗
µ ∈ Q
ω∗
Q
Q (C (G)∗ , τ ω∗ ) F Rx : x ∈ G ∪ Lx : x ∈ G
Q
F µ ≡ Φ(µ)
F =
Lx = ΦLx Φ−1 |Q
Rx F µ (f ) = F µ (f (x ·))
Q F
Rx = ΦRx Φ−1 |Q
Lx F µ (f ) = F µ (f (· x))
f ∈ C (G)
F
Q
S = F = Rx Ly : x, y ∈ G
Rx
Ly
Rx Ry = Ryx Ls Lt = Lst
S
Q
Q
S
F ∈ Q ⊂ C (G)∗ ρ(x, y) = (Rx Ly )(F )
G ρ : G × G → (C (G)∗ , τ ω∗ ) f (x · y)
G
(x, y) → F (f (x · y)) z → f (xzy) f ∈ C (G)
F
(x, y) → f (x · y) f ∈ C (G)
f ∈ C (G) C (G)
F
x, y ∈ G
ε> 0 U x U y x y |f (xzy) − f (xzy)| < ε x ∈ U x y ∈ U y z ∈G ε>0 V e e G w ∈ G |f (w) − f (w)| < ε w ∈ V wV ε > 0 f w ∈ G U w w ε w ∈ U w φ : G×G → G |f (w) − f (w)| < 2 φ(u, v) = uwv (e, e) W w e W w wW w ⊂ U w uv (u, v) → V w V w2 ⊂ W w V w ⊂ W w G G {V w wV w }w∈G
n
w1 ,...,wn ∈ G
G=
n
V wi wi V wi
i=1
w ∈ V wr wr V wr
w ∈ U wr
w ∈ G w ∈ V wV w ∈ V V wr wr V wr V ⊂ W wr wr W wr
V =
V wi
i=1
r ∈ {1, 2,...,n}
|f (w) − f (w)| ≤ |f (w ) − f (wr )| + |f (wr ) − f (w)| < ε.
U x = V x
ε > 0 V e U y = yV
x, y ∈ G
z ∈ G |f (w) ˜ − f (xzy)| < ε x ˜ ∈ U x y˜ ∈ U y z ∈ G
w = xzy w ˜ ∈ V wV f |(˜ xz y˜) − f (xzy)| < ε.
S S
Ly
Rx
S ∈ S f ∈ C (G) Rx F k → 0 y ∈ G
Q Q F k → 0 {F k }k F k (f (x·)) = Rx F k (f ) → 0
Rx
S
S I = Rx Ly J 0∈ / {S (I ) − S (J ) : S ∈ S}
(C (G)∗ , τ ω∗ ) ik (F k ) = F k (f ) → 0 f ∈ C (G) Ly
0 ∈ / {S (I ) − S (J ) : S ∈ S} S ∈ S S
I, J ∈ Q I = J
{S (I ) − S (J ) : S ∈ S} = Rx Ly (I ) − Rx Ly (J ) : x, y ∈ G , ρ(x, y) = Rx Ly (I ) − Rx Ly (J ) ∈ C (G)∗ {S (I ) − S (J ) : S ∈ S} C (G)∗ S F ∈ Q F = Φ(µ)
G×G ρ G×G µ∈Q
ν
F µ (f ) =
fdµ = 1
f ∈ C (G)
G
f (x)dµ(x)dν (y)
=2
f (yx)dµ(x)dν (y) =3
=4
f (y)dν (y)dµ(x) =5
f (yx)dν (y)dµ(x)
f dν = F ν (f ).
µ(G) = ν (G) = 1 f G
