For the convenience of discussion in the sequel we distinguish
the same spaces
L
(
α
)
2
(0
,
1)
,
α
= 1
,
2
, . . . , n
, and construct a product
space
L
n
2
(0
,
1) =
L
(1)
2
(0
,
1)
×
L
(2)
2
(0
,
1)
×
. . .
×
L
(
n
)
2
(0
,
1)
,
L
n
2
(0
,
1) =
=
β
a
β
n
α
=1
ψ
α
β
,
ψ
α
β
∈
L
(
α
)
2
(0
,
1)
,
α
= 1
,
2
, . . . , n
,
β
= 1
,
2
, . . . , N
.
After introduction of inner product for
ψ
,
ϕ
∈
L
n
2
(0
,
1)
,
ψ
=
n
α
=1
ψ
α
,
ϕ
=
n
α
=1
ϕ
α
,
ψ, ϕ
=
n
α
=1
ψ
α
,
n
α
=1
ϕ
α
=
n
α
=1
ψ
α
, ϕ
α
=
n
α
=1
1
0
ψ
α
(
x
α
)
ϕ
α
(
x
α
)
dx
α
,
with induced natural norm
n
α
=1
ψ
α
L
n
2
(0
,
1)
=
n
α
=1
ψ
α
L
(
α
)
2
(0
,
1)
,
L
n
2
(0
,
1)
becomes a linear normed space. The
L
n
2
(0
,
1)
defined above can
be embedded into
L
2
(Ω)
with preserved norm and becomes a dense subset
of the latter [3, 4], while
L
(
α
)
2
(0
,
1)
is a closed subspace of
L
2
(Ω)
, since
ψ
α
=
L
(
α
)
2
(0
,
1)
ψ
α
L
2
(Ω)
always holds on
Ω
the other hand, for the multi-
linear functional
f
:
L
n
2
(0
,
1)
→
R
, defined by
f
n
α
=1
ψ
α
=
Ω
F
(
x
)
α
ψ
α
(
x
α
)
d
Ω
,
(5)
the following inequality holds for every element of
L
n
2
(0
,
1)
:
f
α
ψ
α
≤
M
α
ψ
α
L
(
α
)
2
(0
,
1)
,
where
M
is the lower bound defined in (1). Hence
f
is bounded and, by
Banach-Steinhaus theorem, is totally continuous in
L
n
2
(0
,
1)
[5].
Let
B
α
be the unit ball of
L
(
α
)
2
(0
,
1)
,
B
α
=
{
ψ
α
∈
L
(
α
)
2
(0
,
1)
,
ψ
α
≤
1
}
, and
B
n
=
B
1
×
B
2
× · · ·
B
n
. First of all we need the
following Lemma.
Lemma 1.
The
n
-linear form (5) can achieve its supremum on
B
n
.
Whenever
F
(
x
) = 0
, the supremum
λ
is positive,
λ
= sup
ψ
α
∈
B
α
Ω
F
(
x
)
α
ψ
α
d
Ω
>
0
.
(6)
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
35
