We should note here that ru(A) taken as a function of u defined on a bounded ... structure. It is already known that uniformly convex Banach spaces have uni-.
(α)
! " #$ % & ' !
" #!
' ( ) & ) ) (α)' * * +
$ * ! ,$ * * (α)' * * ) *
! " $ % !!& ' -./0 ()* !' 1 1 (α)' *1 ,$ 1 k '2 + 1 1 $ !
0 2 X # $ , C C $ X ! T : C −→ C (α)' * ) a b a + 2b = 1 30 04 T x − T y ≤ a x − y + b T x − y + T y − x , ∀x, y ∈ C. 5 * ) 607 ! )
$ * (α)' * 8 & b = 0 30 04 ) ) * $ 9 5& T : [0, 1] −→ [0, 1] * * 0 x = 1 T (x) = x = 1 ( 8 $
R (α)' * ( 8 & a=b= . : * T (t) : t ∈ R C X * * C ' * ) * 9 34 T (0) = I ) I * X 34 T (s + t) = T (s)T (t) s, t ∈ R 34 lim T (t)x − x = 0 x ∈ C 607 ) ) * 9 + 607 H C T
T
T
T
T
T
T
1 2
+
1 3
0
+
t→0+
H T (α) C x0 ∈ C T i x0 : i ∈ N x ∈ C 1 i T x, n i=1 n−1
Sn (x) =
T C
607
Γ = T (t) : t ∈ R+ C0 (α) H !
a b t
x0 ∈ C T (t)x0 : t ∈ R+ x ∈ C +
1 St (x) = t
0
t
T (τ )x0 dτ,
H Γ
! * ) * ,$ * 6;7 6 9
> K X ) diam(K) sup x − y : x, y ∈ K .
5 u ∈ A )
r(A) ≤ ru (A) ≤ diam(A).
? r (A) & u , $ A X r : A −→ R ' $ ) C(A) $ ! u ∈ A r (A) = diam(A) u ( $ A ! $ K # X $ H K diam(H) > 0 ! # X $ X 5 8 # X 9 r(K) N(X) = sup : K ⊂ X $ diam(K) > 0 . diam(K) N(X) ≤ 1 N(X) < 1 X ( & ) $ # ' √ ) & ) / 8 = 2 ! * N(X) & ) 8 & ) u
u
u
)
) $ X δ (1) > 0 ! ) = 1 3; 04 λ δ (1 − )N(X) = 1. λ = λ > 1 2 ) G R * N ) x ∈ X x X x x 3; ;4 r(x, {x }) = lim sup x − x. > x ,$ 3; ;4 ,
* $ x X R ) K {x } K X
2 −1 X
−1
+
u u∈G
u u∈G
u
u u∈G
u→+∞
u
+
u u∈G
5
r(K, {xu }) = inf r(x, {xu }) : x ∈ K . xu K A(K, xu ) = x ∈ K : r(x, xu ) = r(K, xu ) . X A(K, xu ) X A(K, xu )
( & ) @$ $ $ * K X ) ) co(K) $ K co(K) = ∩{H : H $ K ⊂ H} ) * ) ) ) * , 67 !
X " {xt }t∈G X y ∈ co({xt : t ∈ G})
3
