class="bi x0 y0 w1 h1"
class="bi x0 y0 w1 h1"
L
1
(Ω; E)
X
∗
L
1
(Ω; E)
X
∗
class="bi x1 yb w2 h7"
L
1
(Ω; E)
F : X ⇒ Y X Y
R
¯
R := R ∪ {±∞}
Q
N
X
∗
X
x
∗
, x X
∗
X
x x

F en
f(x) f x
x
∗
k
w
∗
→ x
∗
{x
∗
k
} x
∗ ∗
w
∗
x := y x y
✷
class="bi x1 y3c w2 hf"
f : [a, b] → R a, b ∈ R

b
a
f

(t)dt = f(b) − f(a)
f

(·)
∂

∈ R
(j ∈ J)
0 ∈ ∂
Cl
x
L(¯x, λ, µ)
λ
i
g
i
(¯x) = 0 ∀i ∈ I,
L(x, λ, µ) := λ
0
f(x) +

i∈I
λ
i
g
i
(x) +

j∈J
µ
j
h
j
(x)
(P) ∂
Cl

x
L(¯x, λ, µ) ⊂ λ
0
∂
Cl
f(¯x) +

i∈I
λ
i
∂
Cl
g
i
(¯x) +

j∈J
µ
j
∂
Cl
h
j
(¯x).
0 ∈ λ
0
∂
Cl
f(¯x) +


i
∂g
i
(¯x) +

j∈J
∂(µ
j
h
j
)(¯x).
0 ∈ λ
0
∂f(¯x) +

i∈I
λ
i
∂g
i
(¯x) +

j∈J
∂(µ
j
h
j
)(¯x).
(P)
G(·)

X
∗
B
X
B
X
∗
G: X ⇒ X
∗
Lim sup
u→x
G(x) :=

x
∗
∈ X
∗



∃u
k
→ x, x
∗
k
w
∗
−→ x
∗
,

t → t
0
t ≥ t
0
f
x ∈ X  > 0 f
x U x
|f(x
1
) −f(x
2
)| ≤ x
1
− x
2
 ∀x
1
, x
2
∈ U.
v ∈ X f x
f
0
(x; v) := lim sup
x

→x, t→0
+
f(x


(u; v) (x, v) v → f
0
(x; v)
X 
(iii) ∂
Cl
f(x)
∗
X
∗
ξ
∗
   ξ
∗
∈ ∂
Cl
f(x);
(iv) v ∈ X, f
0
(x; v) = max{ξ
∗
, v | ξ
∗
∈ ∂
Cl
f(x)};
(v) X = R
n
∂
Cl

F (x) ⊂ W U x F (u) ⊂ W
u ∈ U.
v ∈ X f x
f

(x; v) := lim
t→0
+
f(x + tv) − f(x)
t
,
f
x ∈ X. f x v ∈ X
f

(x; v) f

(x; v) = f
0
(x; v).
f : R → R f(0) = 0 f(x) = x
2
sin
1
x
x ∈ R\{0} 0
0.
(Ω, A, µ)
U X g
ω

(v)dµ(ω)
v
0
∈ U. F U
∂
Cl
F (v) ⊂

Ω
∂
Cl
g
ω
(v)dµ(ω) ∀v ∈ X.
ω ∈ Ω g
ω
(·) v, F
v (1.1)

Ω
∂
Cl
g
ω
(v)dµ(ω)
ξ
∗
∈

Ω

ξ
∗
ω
, xdµ(ω).
ε ≥ 0, ε f
x ∈ X f(x) ∈ R

∂
ε
f(x) :=

x
∗
∈ X
∗



lim inf
u→x
f(u) −f(x) −x
∗
, u −x
u −x
≥ −ε

.
|f(x)| = ∞

∂

∂
ε
k
f(u
k
) x
∗
k
w
∗
→ x
∗
.

∂f(x)
∗
∂f(x)

∂f(x) ⊂ ∂f(x).
f g
X
¯
R, x. f x,

∂(f + g)(x) = f

(x) +

∂g(x).
g = 0

∂f(x) = ∂
F en
f(x).
f : X →
¯
R x ∈ X
f(x)  lim inf
u→x
f(u) lim inf
u→x
f(u) := sup
U∈N (x)
inf
u∈U
f(u) N(x)
X x f
x U ∈ N(x) f u ∈ U
X X
X
f : U → R
U ⊂ X U

1
C[0, 1] L
1
[0, 1]
B
X
∗
X

∂f(x) = Lim sup
f
u → x

∂f(u);
(iii) (f
1
, f
2
) ∈ LS(x) ε  0 γ > 0

∂
ε
(f
1
+ f
2
)(x) ⊂



∂f
1
(x
1
) +

∂f
2
(x

f(x)
∂f(x)
(Ω, A, µ) σ− G : Ω ⇒ R
n
Ω R
n
G
G
−1
(W ) := {ω ∈ Ω | G(ω) ∩ W = ∅} ∈ A
W ⊂ R
n
G k(·) ∈ L
1
(Ω)
G(ω) ⊂ k(ω)B
R
n
Ω L
1
(Ω)
Ω R
G : Ω ⇒ R
n
, Ω
R
n
,
G
G G,

Ω
g
n
dµ

g = (g
1
, , g
n
).

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