Proof.
Since all unit balls
B
α
,
α
= 1
,
2
, . . . , n
, are weak compact,
by the Banach-Alaoglu theorem, every sequence
{
ψ
α
k
}
enabling (4) to
approach supremum contains a subsequence weakly converging to some
element
ϕ
α
in
B
α
[5, 6]. Now we show that there exists a sequence
n
α
=1
ψ
α
k
,
k
= 1
,
2
, . . .
, that converges weakly to an element
n
α
=1
ϕ
α
∈
L
n
2
(0
,
1)
at
which the following functional achieves its supremum,
lim
k
→∞
Ω
F
(
x
1
,
· · ·
, x
n
)
α
ψ
α
k
d
Ω =
=
Ω
F
(
x
1
,
· · ·
, x
n
)
α
ϕ
α
d
Ω =
λ
= sup
ψ
α
∈
B
α
Ω
F
(
x
)
α
ψ
α
(
x
α
)
d
Ω
,
or
lim
k
→∞
Ω
F
(
x
1
,
· · ·
, x
n
)[
α
ψ
α
k
−
α
ϕ
α
]
d
Ω = 0
.
Due to the identity
α
ψ
α
k
−
α
ϕ
α
=
=
α
ψ
α
k
−
ϕ
1
ψ
2
k
· · ·
ψ
n
k
+
ϕ
1
ψ
2
k
· · ·
ψ
n
k
−
ϕ
1
ϕ
2
ψ
3
k
· · ·
ψ
n
k
+
· · ·
+
ϕ
1
ϕ
2
· · ·
ϕ
n
−
1
ψ
n
k
−
α
ϕ
α
,
we have
Ω
F
α
ψ
α
k
−
α
ϕ
α
d
Ω =
=
n
i
=1 Ω
i
1
0
F
·
(
ψ
i
k
−
ϕ
i
)
dx
i
i
−
1
α
=1
ϕ
α
n
β
=
i
+1
ψ
β
k
d
Ω
i
=
=
n
i
=1 Ω
i
F
k
i
−
1
α
=1
ϕ
α
·
n
β
=
i
+1
ψ
β
k
d
Ω
i
,
where
d
Ω
i
=
dx
1
· · ·
dx
ˆ
i
· · ·
dx
n
;
ˆ
i
means without ith coordinate, and
F
k
(
x
1
,
· · ·
, x
ˆ
i
,
· · ·
, x
n
) =
1
0
F
(
x
1
,
· · ·
, x
n
)[
ψ
i
k
(
x
i
)
−
ϕ
i
(
x
i
)]
dx
i
.
36
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
