# The Tensor Product Representation of Polynomials of Weak Type in a DF-Space

Authors: Masaru Nishihara; Kwang Ho Shon
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetE and F be locally convex spaces over C and let P
(
n
E
;
F
) be the space of all continuous n-homogeneous polynomials from E to F. We denote by ⨂
n
,
s
,
π
E the n-fold symmetric tensor product space of E endowed with the projective topology. Then, it is well known that each polynomial p
∈
P
(
n
E
;
F
) is represented as an element in the space L
(
⨂
n
,
s
,
π
E
;
F
) of all continuous linear mappings from ⨂
n
,
s
,
π
E to F. A polynomial p
∈
P
(
n
E
;
F
) is said to be of weak type if, for every bounded set B of E, p
|
B is weakly continuous on B. We denote by P
w
(
n
E
;
F
) the space of all n-homogeneous polynomials of weak type from E to F. In this paper, in case that E is a DF space, we will give the tensor product representation of the space P
w
(
n
E
;
F
).


---

## Body

## 1. Notations and Preliminaries

In this section, we collect some notations, some definitions, and some basic properties of locally convex spaces which we use throughout this paper.LetE
1 and E
2 be complex vector spaces. Then the pair 〈
E
1
,
E
2
〉 is called adual pair if there exists a bilinear form:
(1)
(
x
1
,
x
2
)
⟶
〈
x
1
,
x
2
〉
(
(
x
1
,
x
2
)
∈
E
1
×
E
2
)
satisfying the following conditions: (1)
If〈
x
1
,
x
2
〉
=
0 for every x
2
∈
E
2, x
1
=
0.
(2)
If〈
x
1
,
x
2
〉
=
0 for every x
1
∈
E
1, x
2
=
0.We denote byσ
(
E
1
,
E
2
) (resp., σ
(
E
2
,
E
1
)) the topology on E
1 (resp., E
2) defined by the subset of seminorms:
(2)
{
|
〈
·
,
x
2
〉
|
;
x
2
∈
E
2
}
(
resp
.
{
|
〈
x
1
,
·
〉
|
;
x
1
∈
E
1
}
)
.LetE be a locally convex space. We denote by c
s
(
E
) the set of all nontrivial continuous seminorms on E. The topology σ
(
E
,
E
′
) on E is called theweak topology of E and the topology σ
(
E
′
,
E
) on E
′ is called theweak
*
topology of E
′. We denote by ℬ
(
E
) the family of all bounded subsets of E. We denote by ∥
∥
B the seminorm on E
′ defined by
(3)
∥
x
′
∥
B
=
sup
⁡
{
|
〈
x
,
x
′
〉
|
;
x
∈
B
}
,
for every B
∈
ℬ
(
E
). Thestrong topology on E
′ is the topology on E
′ defined by the set of seminorms {
∥
∥
B
;
B
∈
ℬ
(
E
)
} on E
′. We denote by E
β
′ the locally convex space E
′ endowed with the strong topology. We denote by E
′′ the dual space of E
β
′. Let A be a subset of E. We denote by A
¯ the topological closure of the subset A of E
⊂
E
′′ for the topology σ
(
E
′′
,
E
′
). Thepolar set
A
∘ of A is defined by
(4)
A
∘
=
{
x
′
∈
E
′
;
sup
⁡
x
∈
A
|
〈
x
,
x
′
〉
|
≤
1
}
.We denote byA
∘
∘ the bipolar set of A for the dual pair 〈
E
′′
,
E
′
〉. A subset S of E
′ is said to beequicontinuous if there exists a neighborhood V of 0 in E such that S
⊂
V
∘.Lemma 1.
LetM be a bounded subset of a locally convex space E. Then the following statements hold:(1)
M
¯
=
M
∘
∘ if M is absolutely convex.
(2)
M
¯ is compact with the topology σ
(
E
′′
,
E
′
).
(3)
E
′′
=
⋃
M
∈
ℬ
(
E
)
M
¯.
(4)
LetA be an equicontinuous subset of E
′′ for the dual pair 〈
E
′′
,
E
′
〉. Then there exists an absolutely convex bounded subset M of E, such that A
⊂
M
∘
∘.Proof.
(
1
) Since M
⊂
M
∘
∘ and M
∘
∘ is σ
(
E
′′
,
E
′
)-closed, M
¯
⊂
M
∘
∘. We shall show M
∘
∘
⊂
M
¯. We assume that x
0
′′
∉
M
¯. We denote by E
σ
(
E
′′
,
E
′
)
′′ the locally convex space E
′′ endowed with the topology σ
(
E
′′
,
E
′
). Since (
E
σ
(
E
′′
,
E
′
)
′′
)
′
=
E
′ and M
¯ is σ
(
E
′′
,
E
′
)-closed absolutely convex, by Hahn-Banach theorem there exists x
′
∈
E
′ such that
(5)
sup
⁡
x
′′
∈
M
¯
|
〈
x
′′
,
x
′
〉
|
≤
1
,
|
〈
x
0
′′
,
x
′
〉
|
>
1
.
Thus, it is valid that x
′
∈
M
¯
∘ and x
0
′′
∉
M
¯
∘
∘. Thus we have M
¯
∘
∘
⊂
M
¯. Since M
∘
∘
⊂
M
¯
∘
∘, M
∘
∘
⊂
M
¯. Hence, we have M
¯
=
M
∘
∘.
(
2
) We denote by Γ
(
M
) the σ
(
E
′′
,
E
′
)-closed absolutely convex hull of the set M. By the statement (
1
) we have
(6)
Γ
(
M
)
=
M
∘
∘
.
The polar set M
∘ is an absolutely convex neighborhood of 0 in E
β
′. Therefore, Γ
(
M
) is a σ
(
E
′′
,
E
′
)-closed equicontinuous subset of E
′′. By Banach-Alaoglu theorem, Γ
(
M
) is σ
(
E
′′
,
E
′
)-compact. Since M
¯
⊂
Γ
(
M
), M
¯ is also σ
(
E
′′
,
E
′
)-compact.
(
3
) It is clear that ⋃
M
∈
ℬ
(
E
)
M
¯
⊂
E
′′. We shall show that E
′′
⊂
⋃
M
∈
ℬ
(
E
)
M
¯. Let x
′′ be a point of E
′′. Then, there exists an open neighborhood V of 0 in E
β
′ such that |
〈
x
,
x
′′
〉
|
≤
1 for every x
∈
V. By the definition of the space E
β
′ there exists an absolutely convex bounded subset M
∈
ℬ
(
E
) such that M
∘
⊂
V. Thus, by the statement (
1
) we have
(7)
x
′′
∈
V
∘
⊂
M
∘
∘
=
M
¯
.
Thus, we have E
′′
=
⋃
M
∈
ℬ
(
E
)
M
¯.
(
4
) Since A is an equicontinuous subset of E
′′, there exists a neighborhood V of 0 in E
β
′ such that A
⊂
V
∘. Since V is a neighborhood of 0 in E
β
′, there exists an absolutely convex bounded subset M of E such that M
∘
⊂
V. Thus, we have A
⊂
V
∘
⊂
M
∘
∘. This completes the proof.A filterℱ
=
{
F
α
} of a locally convex space E is called aCauchy filter if for every neighborhood U of 0 there exists an F
∈
ℱ such that {
x
-
y
;
x
,
y
∈
F
}
⊂
U. A locally convex space E is said to becomplete if any Cauchy filter on E converges to a point of E. There exists the smallest complete locally convex space E
~ containing E as a subspace. The locally convex space E
~ is called thecompletion of E.


## 2. The Extension of Polynomial Mappings of Weak Type

In this section, we will give basic properties of polynomial mappings on locally convex spaces and discuss the extension of weak type on locally convex spaces. For more detailed properties of polynomials on locally convex spaces, see Dineen [1, 2] and Mujica [3]. Let E and F be locally convex spaces and let n be a positive integer. We denote by L
a
(
n
E
;
F
) the space of all n-linear mappings from the product space E
n of n-copies of E into F and denote by L
a
w
(
n
E
;
F
) the space of all n-linear mappings, which are σ
(
E
,
E
′
)-continuous on bounded subsets of E
n, from the product space E
n into F. A mapping p
:
E
→
F is called an n-homogeneous polynomial from E into F if there exists an n-linear mapping u from E into F such that
(8)
P
(
x
)
=
u
(
x
,
…
,
x
)
,
for every x
∈
E. If p is an n-homogeneous polynomial from E into F, there exists uniquely a symmetric n-linear mapping u. We denote by P
a
(
n
E
;
F
) the space of all n-homogeneous polynomials from E into F. We denote by P
(
n
E
;
F
) (resp., L
(
n
E
;
F
)) the space of all continuous n-homogeneous polynomials from E (resp., all continuous n-linear mappings from E
n) into F. We denote by P
a
w
(
n
E
;
F
) the space of all σ
(
E
,
E
′
)-continuous polynomials on each bounded subset of E. We set
(9)
P
w
(
n
E
;
F
)
=
P
(
n
E
;
F
)
∩
P
a
w
(
n
E
;
F
)
,
L
w
(
n
E
;
F
)
=
L
(
n
E
;
F
)
∩
L
a
w
(
n
E
;
F
)
.A polynomial belonging toP
w
(
n
E
;
F
) is said to beof weak type.Lemma 2.
LetE and F be locally convex spaces and let u be an n-linear mapping belonging to L
w
(
n
E
;
F
). Let A
1
,
…
,
A
n be absolutely convex bounded subsets of E. Let a
i be any point of σ
(
E
′′
,
E
′
)-closure A
¯
i of A
i for each i with 2
≤
i
≤
n. We denote by 𝒩
w
*
(
E
′′
)
(
0
) the system of all σ
(
E
′′
,
E
′
)-neighborhoods of 0 in E
′′. Then, for any α
∈
c
s
(
F
) there exists V
∈
𝒩
w
*
(
E
′′
)
(
0
) such that
(10)
α
(
u
(
x
1
,
…
,
x
n
)
)
<
1
,
for every (
x
1
,
…
,
x
n
)
∈
(
V
∩
A
1
)
×
(
(
a
2
+
V
)
∩
A
2
)
×
⋯
×
(
(
a
n
+
V
)
∩
A
n
).Proof.
We shall prove this lemma by induction onn. Let n
=
1. Then, the conclusion of this lemma is true since 0
∈
A
1 and σ
(
E
,
E
′
) is the induced topology of the topology σ
(
E
′′
,
E
′
) onto E.
We suppose that the conclusion of this lemma is true for all mappings belonging toL
w
(
n
-
1
E
;
F
). And we assume that for u
∈
L
w
(
n
E
;
F
), the conclusion is not true. Then, there exists α
∈
c
s
(
F
) such that for every V
∈
𝒩
w
*
(
E
′′
)
(
0
) there is a point:
(11)
(
x
1
V
,
x
2
V
,
…
,
x
n
V
)
∈
(
V
∩
A
1
)
×
(
(
a
2
+
V
)
∩
A
2
)
×
⋯
×
(
(
a
n
+
V
)
∩
A
n
)
,
satisfying
(12)
α
(
u
(
x
1
V
,
x
2
V
,
…
,
x
n
V
)
)
≥
1
.
Sinceu is σ
(
E
,
E
′
)-continuous at (
0,0
,
…
,
0
) on each absolutely convex bounded subset of E
n and σ
(
E
,
E
′
) is the induced topology of σ
(
E
′′
,
E
′
) onto E, there exists W
∈
𝒩
w
*
(
E
′′
)
(
0
) such that
(13)
α
(
u
(
x
1
,
x
2
,
…
,
x
n
)
)
≤
1
4
for every x
i
∈
2
(
W
∩
A
i
) with 1
≤
i
≤
n. We choose a decreasing sequence W
⊃
W
(
1
)
⊃
W
(
2
)
⊃
⋯
⊃
W
(
n
-
1
) of elements of 𝒩
w
*
(
E
′′
)
(
0
) by the process of (
n
-
1
) steps as follows.
At the first step, we consider the(
n
-
1
)-linear mapping:
(14)
(
z
1
,
…
,
z
n
-
1
)
⟶
u
(
z
1
,
…
,
z
n
-
1
,
x
n
W
)
,
belonging to L
w
(
n
-
1
E
;
F
). By the assumption of induction, there exists W
(
1
)
∈
𝒩
w
*
(
E
′′
)
(
0
) with W
(
1
)
⊂
W such that
(15)
α
(
u
(
z
1
,
…
,
z
n
-
1
,
x
n
W
)
)
≤
1
2
n
for every (
z
1
,
…
,
z
n
-
1
)
∈
(
W
(
1
)
∩
A
1
)
×
(
(
a
2
+
W
(
1
)
)
∩
A
2
)
×
⋯
×
(
(
a
n
-
1
+
W
(
1
)
)
∩
A
n
-
1
).
At the second step, we consider the(
n
-
1
)-linear mapping
(16)
(
z
1
,
…
,
z
n
-
2
,
z
n
)
⟶
u
(
z
1
,
…
,
z
n
-
2
,
x
n
-
1
W
(
1
)
,
z
n
)
belonging to L
w
(
n
-
1
E
;
F
). By the assumption of induction, there exists W
(
2
)
∈
𝒩
w
*
(
E
′′
)
(
0
) with W
(
2
)
⊂
W
(
1
) such that
(17)
α
(
u
(
z
1
,
…
,
z
n
-
2
,
x
n
-
1
W
(
1
)
,
z
n
)
)
≤
1
2
n
for every point (
z
1
,
…
,
z
n
-
2
,
z
n
) of
(18)
(
W
(
2
)
∩
A
1
)
×
(
(
a
2
+
W
(
2
)
)
∩
A
2
)
×
⋯
×
(
(
a
n
-
2
+
W
(
2
)
)
∩
A
n
-
2
)
×
(
(
a
n
+
W
(
2
)
)
∩
A
n
)
.
Repeating this process, at the(
n
-
1
)th step, we consider the (
n
-
1
)-linear mapping:
(19)
(
z
1
,
z
3
,
…
,
z
n
)
⟶
u
(
z
1
,
x
2
W
(
n
-
2
)
,
z
3
,
…
,
z
n
)
,
belonging to L
w
(
n
-
1
E
;
F
). By the assumption of induction, there exists W
(
n
-
1
)
∈
𝒩
w
*
(
E
′′
)
(
0
) with W
(
n
-
1
)
⊂
W
(
n
-
2
) such that
(20)
α
(
u
(
z
1
,
x
2
W
(
n
-
2
)
,
z
3
,
…
,
z
n
)
)
≤
1
2
n
for every (
z
1
,
z
3
…
,
z
n
)
∈
(
W
(
n
-
1
)
∩
A
1
)
×
(
(
a
3
+
W
(
n
-
1
)
)
∩
A
3
)
×
⋯
×
(
(
a
n
+
W
(
n
-
1
)
)
∩
A
n
). Then we have
(21)
α
(
u
(
x
1
W
(
n
-
1
)
,
x
2
W
(
n
-
1
)
-
x
2
W
(
n
-
2
)
,
…
,
x
n
-
1
W
(
n
-
1
)
mmm
-
x
n
-
1
W
(
1
)
,
x
n
W
(
n
-
1
)
-
x
n
W
)
)
≥
α
(
u
(
x
1
W
(
n
-
1
)
,
x
2
W
(
n
-
1
)
,
…
,
x
n
W
(
n
-
1
)
)
)
-
∑
(
k
i
)
∈
K
‍
α
(
u
(
x
1
W
(
n
-
1
)
,
x
2
W
(
k
2
)
,
…
,
x
n
W
(
k
n
)
)
)
≥
1
-
(
2
n
-
1
-
1
)
×
1
2
n
≥
1
2
,
where K
=
{
(
k
2
,
…
,
k
n
)
;
k
i
∈
{
n
-
1
,
n
-
i
}
,
i
=
2
,
…
,
n
}
∖
{
(
n
-
1
,
…
,
n
-
1
)
} and W
(
0
)
=
W.
If we set(22)
(
y
1
,
y
2
,
…
,
y
n
)
=
(
x
1
W
(
n
-
1
)
,
x
2
W
(
n
-
1
)
-
x
2
W
(
n
-
2
)
,
…
,
x
n
W
(
n
-
1
)
-
x
n
W
)
,
we have
(23)
α
(
u
(
y
1
,
…
,
y
n
)
)
≥
1
2
,
(
y
1
,
y
2
,
…
,
y
n
)
∈
(
W
∩
A
1
)
×
{
2
(
W
∩
A
2
)
}
×
⋯
×
{
2
(
W
∩
A
n
)
}
.
This is a contradiction by (13). This completes the proof.Lemma 3.
LetE be a locally convex space, let F be a complete locally convex space and let u be an n-linear mapping belonging to L
w
(
n
E
;
F
). Let A
1
,
…
,
A
n be absolutely convex bounded subsets of E. Then there exists a σ
(
E
′′
,
E
′
)-continuous mapping u
~
A
¯
1
×
⋯
×
A
¯
n from A
¯
1
×
⋯
×
A
¯
n into F such that u
~
A
¯
1
×
⋯
×
A
¯
n
=
u on A
1
×
⋯
×
A
n.Proof.
Leta
i be any point of the σ
(
E
′′
,
E
′
)-closure A
¯
i of A
i for each i with 1
≤
i
≤
n. At first, we shall show that a filter of F
(24)
ℱ
(
a
1
,
…
,
a
n
)
=
{
V
∈
𝒩
w
*
(
E
′′
)
u
(
(
a
1
+
V
)
∩
A
1
,
…
,
(
a
n
+
V
)
∩
A
n
)
;
V
∈
𝒩
w
*
(
E
′′
)
(
0
)
}
is a Cauchy filter. Let α be an arbitrary continuous seminorm of F. By Lemma 2 there exists V
∈
𝒩
w
*
(
E
′′
)
(
0
) such that
(25)
α
(
u
(
x
1
,
x
2
,
…
,
x
n
)
)
<
1
n
,
for every point (
x
1
,
x
2
,
…
,
x
n
) of
(26)
⋃
i
=
1
n
‍
(
∏
j
=
1
n
‍
(
(
γ
i
j
a
j
+
2
V
)
∩
2
A
j
)
)
,
where γ
i
j
=
1
(
i
≠
j
) and γ
i
j
=
0
(
i
=
j
). Then, we have
(27)
α
(
u
(
x
1
,
x
2
,
…
,
x
n
)
-
u
(
y
1
,
y
2
,
…
,
y
n
)
)
≤
α
(
u
(
x
1
-
y
1
,
x
2
,
…
,
x
n
)
)
+
α
(
u
(
y
1
,
x
2
-
y
2
,
x
3
,
…
,
x
n
)
)
+
α
(
u
(
y
1
,
y
2
,
x
3
-
y
3
,
x
4
,
…
,
x
n
)
)
+
⋯
+
α
(
u
(
y
1
,
y
2
,
…
,
y
n
-
1
,
x
n
-
y
n
)
)
≤
n
×
1
n
=
1
,
for every (
x
1
,
…
,
x
n
)
,
(
y
1
,
…
,
y
n
)
∈
∏
i
=
1
n
‍
(
(
a
i
+
V
)
∩
A
i
). Thus, the filter ℱ
(
a
1
,
…
,
a
n
) is a Cauchy filter. Since F is complete and Hausdorff, there exists uniquely the limit point of the filter ℱ
(
a
1
,
…
,
a
n
) for every (
a
1
,
…
,
a
n
)
∈
A
¯
1
×
⋯
×
A
¯
n. We denote by u
~
A
¯
1
×
⋯
×
A
¯
n
(
a
1
,
…
,
a
n
) the limit point of the filter ℱ
(
a
1
,
…
,
a
n
) for every (
a
1
,
…
,
a
n
)
∈
A
¯
1
×
⋯
×
A
¯
n. Then u
~
A
¯
1
×
⋯
×
A
¯
n defines σ
(
E
′′
,
E
′
)-continuous mapping from A
¯
1
×
⋯
×
A
¯
n into F with
(28)
u
~
A
¯
1
×
⋯
×
A
¯
n
=
u
on
⁡
A
1
×
⋯
×
A
n
.
This completes the proof.Lemma 4.
LetE be a locally convex space, let F be a complete locally convex space, and let u be an n-linear mapping belonging to L
w
(
n
E
;
F
). Then, there exists u
~
∈
L
a
(
n
E
′′
;
F
) with u
~
∣
E
n
=
u such that u
~ is σ
(
E
′′
,
E
′
)-continuous on A
¯
1
×
⋯
×
A
¯
n for all absolutely convex bounded subsets A
1
,
…
,
A
n of E.Proof.
By Lemma3 for all absolutely convex bounded subsets A
1
,
…
,
A
n of E, there exists a σ
(
E
′′
,
E
′
)-continuous mapping u
~
A
¯
1
×
⋯
×
A
¯
n from A
¯
1
×
⋯
×
A
¯
n to F with u
~
A
¯
1
×
⋯
×
A
¯
n
=
u on A
1
×
⋯
×
A
n. If A
1
,
…
,
A
n
,
B
1
,
…
,
B
n are absolutely convex bounded subsets of E with A
¯
i
∩
B
¯
i
≠
∅ for 1
≤
i
≤
n, we have
(29)
u
~
A
¯
1
×
⋯
×
A
¯
n
=
u
~
B
¯
1
×
⋯
×
B
¯
n
on (
A
¯
1
×
⋯
×
A
¯
n
)
∩
(
B
¯
1
×
⋯
×
B
¯
n
). Thus, by Lemma 1, we can define ann-linear mapping u
~ from E
′′
n into F by setting u
~
=
u
~
A
¯
1
×
⋯
×
A
¯
n on A
¯
1
×
⋯
×
A
¯
n for all absolutely convex bounded subsets A
1
,
…
,
A
n of E. Then, the mapping u
~ satisfies all required conditions of this lemma. This completes the proof.The following theorem is proved by Aron et al. [4], González and Gutiérrez [5], Honda et al. [6].Theorem 5.
LetE be a locally convex space and let F be a complete locally convex space. Let p
∈
P
(
n
E
;
F
). Then, the following statements are equivalent. (1)
p
∈
P
w
(
n
E
;
F
).
(2)
For each absolutely convex bounded subsetM of E, p can be extended σ
(
E
′′
,
E
′
)-continuously to M
∘
∘, where M
∘
∘ is the bipolar set of M for the dual pair 〈
E
′′
,
E
′
〉.
(3)
There existsp
~
∈
P
a
(
n
E
′′
;
F
) such that p
~ is σ
(
E
′′
,
E
′
)-continuous on each equicontinuous subset of E
′′ and p
~
∣
E
=
p.
(4)
p is weakly uniformly continuous on every bounded subset of E.Proof.
We shall show that(
1
) implies (
3
). There exists a symmetric n-linear mapping u from E
n into F such that p
(
x
)
=
u
(
x
,
…
,
x
) for every x
∈
E. By the polarization formula, we have u
∈
L
w
(
n
E
,
F
). By Lemma 4 there exists u
~
∈
L
a
(
n
E
′′
;
F
) with u
~
∣
E
n
=
u such that u
~ is σ
(
E
′′
,
E
′
)-continuous on A
¯
1
×
⋯
×
A
¯
n for all absolutely convex bounded subsets A
1
,
…
,
A
n of E. We define p
~
∈
P
a
(
n
E
′′
;
F
) by p
~
(
x
)
=
u
~
(
x
,
…
,
x
) for every x
∈
E
′′. By Lemma 1, p
~ satisfies all required conditions of the statement (
3
).
(
3
) implies (
2
) since M
∘
∘ is equicontinuous for every absolutely convex bounded subset M of E.
We shall show that(
2
) implies (
4
). Let A be a bounded subset of E. We denote by M the absolutely convex hull of A in E. Then, M is a bounded subset of E, and A
⊂
M. By statement (
2
), there exists a σ
(
E
′′
,
E
′
)-continuous mapping p
~
M
¯ from M
¯ into F such that p
~
M
¯
=
p on M. Since by Lemma 1  M
¯ is σ
(
E
′′
,
E
′
)-compact, p
~
M
¯ is uniformly σ
(
E
′′
,
E
′
)-continuous on M
¯. Since p
~
M
¯
=
p on A, p is weakly uniformly continuous on A. This implies (
4
). It is clear that (
4
) implies (
1
). This completes the proof.


## 3. The Tensor Product Representation of Polynomials in Locally Convex Spaces

For anyu
∈
L
a
(
n
E
;
F
), we set
(30)
s
(
u
)
(
x
1
,
…
,
x
n
)
=
1
n
!
∑
σ
∈
S
n
‍
u
(
x
σ
(
1
)
,
…
,
x
σ
(
n
)
)
for every (
x
1
,
…
,
x
n
)
∈
E
n, where S
n is the permutation group of degree n. Then, s
(
u
) is a symmetric n-linear mapping from E
n to F satisfying
(31)
u
(
x
,
…
,
x
)
=
s
(
u
)
(
x
,
…
,
x
)
for every x
∈
E. We denote L
a
s
(
n
E
;
F
) by the space of all symmetric n-linear mappings from E
n to F. Let Δ
n be the mapping from E into E
n defined by
(32)
Δ
n
(
x
)
=
(
x
,
…
,
x
)
for
every
x
∈
E
.For anyu
∈
L
a
(
n
E
;
F
), we define an n-homogeneous polynomial Δ
n
*
(
u
) by
(33)
Δ
n
*
(
u
)
=
u
∘
Δ
n
.
The mapping
(34)
Δ
n
*
:
L
a
s
(
n
E
;
F
)
⟶
P
a
(
n
E
;
F
)
is surjective.Theorem 6 (polarization formula).
Letp
∈
P
a
(
n
E
;
F
) and let u
∈
L
a
s
(
n
E
;
F
). If Δ
n
*
(
u
)
=
p, then
(35)
u
(
x
1
,
…
,
x
n
)
=
1
2
n
n
!
∑
ϵ
i
=
±
1
‍
ϵ
1
⋯
ϵ
n
p
(
∑
i
=
1
n
‍
ϵ
i
x
i
)
.By the above polarization formula, the mappingΔ
n
*
:
L
a
s
(
n
E
;
F
)
→
P
a
(
n
E
;
F
) is a linear isomorphism.We denote by⨂
n
E the n-fold tensor product space of E. Let i
n be the linear mapping from E
n into ⨂
n
E defined by
(36)
i
n
(
x
1
,
…
,
x
n
)
=
x
1
⊗
⋯
⊗
x
n
for every (
x
1
,
…
,
x
n
)
∈
E
n. For any u
∈
L
a
(
n
E
;
F
), there exists a unique i
n
*
(
u
)
∈
L
a
(
⨂
n
E
;
F
) such that the diagram
(37)
commutes. The mapping
(38)
i
n
*
:
L
a
(
n
E
;
F
)
⟶
L
a
(
⨂
n
E
;
F
)
is a linear isomorphism. Each element of ⨂
n
E has a representation of the form
(39)
∑
i
=
1
ℓ
‍
x
i
,
1
⊗
x
i
,
2
⊗
⋯
⊗
x
i
,
n
.However, this representation will never be unique. We denote byδ
n the mapping of E into ⨂
n
E defined by
(40)
δ
n
(
x
)
=
x
⊗
⋯
⊗
x
for every x
∈
E.Proposition 7.
A mappingp
:
E
→
F is an n-homogeneous polynomial if and only if there exists T
∈
L
a
(
n
E
;
F
) such that the diagram
(41)
commutes.For anyx
1
⊗
⋯
⊗
x
n
∈
⨂
n
E, we set
(42)
s
(
x
1
⊗
⋯
⊗
x
n
)
=
1
n
!
∑
σ
∈
S
n
‍
x
σ
(
1
)
⊗
⋯
⊗
x
σ
(
n
)
.We denote by⨂
n
,
s
E the subspace of ⨂
n
E generated by s
(
x
1
⊗
⋯
⊗
x
n
)
,
x
i
∈
E. The space ⨂
n
,
s
E is calledthe
n
-fold symmetric tensor product space of E. Elements of ⨂
n
,
s
E are called n
-symmetric tensors. Clearly, every tensor of the form x
⊗
⋯
⊗
x is a symmetric tensor. Moreover, each element θ in ⨂
n
,
s
E can be expressed as a finite sum (not necessarily unique) of the form
(43)
∑
i
‍
x
i
⊗
⋯
⊗
x
i
.
For any p
∈
P
a
(
n
E
;
F
), there exists a unique j
n
*
(
p
)
∈
L
a
(
⨂
n
,
s
E
;
F
) such that the diagram
(44)
commutes. The mapping
(45)
j
n
*
:
P
a
(
n
E
;
F
)
⟶
L
a
(
⨂
n
,
s
E
;
F
)
is a linear isomorphism. Let P
(
n
E
;
F
), L
(
n
E
;
F
), and L
s
(
n
E
;
F
) be, respectively, the spaces of continuous n-homogeneous polynomials from E into F and the continuous symmetric n-linear mappings from E into F. The restrictions
(46)
Δ
n
*
:
L
(
n
E
;
F
)
⟶
P
(
n
E
;
F
)
,
Δ
n
*
:
L
s
(
n
E
;
F
)
⟶
P
(
n
E
;
F
)
,
s
:
L
(
n
E
;
F
)
⟶
L
s
(
n
E
;
F
)
are well-defined. For each α
∈
c
s
(
E
) and θ
=
∑
i
‍
x
i
⊗
⋯
⊗
x
i
∈
⨂
n
,
s
E, we set
(47)
π
α
,
n
(
θ
)
=
inf
⁡
{
∑
i
‍
α
(
x
i
)
n
∣
θ
=
∑
i
‍
x
i
⊗
⋯
⊗
x
i
}
.
π
α
,
n is a seminorm on ⨂
n
,
s
E. We define the π-topology or the projective topology on ⨂
n
,
s
E as the locally convex topology generated by {
π
α
,
n
}
α
∈
c
s
(
E
). We denote by ⨂
n
,
s
,
π
E the space endowed with π-topology and denote by ⨂
¯
n
,
s
,
π
E the completion ⨂
n
,
s
,
π
E. Then, the following is valid (cf. Dineen [2]).Proposition 8.
LetE be a locally convex space, then we have
(48)
L
(
n
E
;
F
)
≅
L
(
⨂
n
.
π
E
;
F
)
,
L
s
(
n
E
;
F
)
≅
L
(
⨂
n
.
s
,
π
E
;
F
)
≅
P
(
n
E
;
F
)
.Letℰ
(
E
′′
) be the family of all equicontinuous subsets of E
′′ with respect to the dual pair 〈
E
′′
,
E
′
〉. We denote by 𝒪
w
*
,
ϵ the family of subsets of E
′′ defined by
(49)
𝒪
w
*
,
ϵ
=
{
V
∣
V
⊂
E
′′
,
V
∩
M
are
σ
(
E
′′
,
E
′
)
-open
in
M
for
all
M
∈
ℰ
(
E
′′
)
}
.
We denote by τ
w
*
,
ϵ the topology on E
′′ such that the family of all τ
w
*
,
ϵ-open sets coincides with 𝒪
w
*
,
ϵ. By Theorem 5, the following is valid.Proposition 9.
An-homogeneous polynomial p on E is of weak type if and only if there exists a τ
w
*
,
ϵ-continuous n-homogeneous polynomial p
~ on E
′′ such that p
~
=
p on E.However, in general, the topologyτ
w
*
,
β is not a locally convex topology (cf. Kōmura [7]).Definition 10.
A locally convex spaceE is called a DF-space if it contains a countable fundamental system of bounded sets and if the intersection of any sequence of absolutely convex neighborhoods of 0 which absorbs all bounded sets is itself a neighborhood of 0.Grothendieck [8, 9] proved that the strong dual space of a metrizable locally convex space is a DF-space and the strong dual space of a DF-space is a Fréchet space.All Banach spaces are DF-spaces. The following result is known.Proposition 11 (Banach-Dieudonné theorem).
LetE be a metrizable locally convex space. Then, the topology τ
w
*
,
ϵ on E
′ is the topology of uniform convergence on all compact sets in E.Proof.
The proof is on Köthe [10, § 21–10].IfE is a DF space, then E
β
′ is a Fréchet space. Therefore, by Proposition 11 the following is valid.Proposition 12.
The topologyτ
w
*
,
ϵ on E
′′ is the topology of uniform convergence on all compact sets in E
′.We denote byα
K the seminorm on E
′′ defined by
(50)
α
K
(
x
′′
)
=
sup
⁡
{
|
〈
x
′′
,
x
′
〉
|
∣
x
′
∈
K
}
for every compact subsets K of E
β
′. We denote by τ
0 the locally convex topology on E
′′ defined by the set of seminorms:
(51)
{
α
K
∣
K
is
every
compact
set
of
E
′
}
.
We denote by E
τ
0 the locally convex space E defined the set of seminorms:
(52)
{
α
K
|
E
∣
K
is
every
compact
set
of
E
′
}
.
An n-homogeneous polynomial p on E
′′ is σ
(
E
′′
,
E
′
)-continuous on every equicontinuous subsets of E
′′ if and only if p is τ
w
*
,
ϵ-continuous on E
′′. Thus, by Propositions 9 and 11, we have the following theorem.Theorem 13.
If ann-homogeneous polynomial p on E is of weak type if and only if p is τ
o-continuous in E and continuous on E with the initial topology of E.Then, we can obtain the following topological tensor product representation of polynomials of weak type.Theorem 14.
LetE be a DF space, then we have
(53)
L
(
⨂
n
,
s
π
E
τ
0
;
F
)
∩
L
(
⨂
n
,
s
π
E
;
F
)
≅
P
w
(
n
E
;
F
)
for every complete locally convex space F.A topological spaceX is called a k
-space if its topology is localized on its compact set;, that is, U
⊂
X is open if and only if U
∩
K is open in K, with the induced topology, for each compact subset K of X. A mapping from a k-space into a topological space is continuous if and only if its restriction to each compact set is continuous.Letu be a mapping from a locally convex space E into a locally convex space F. If E is a k-space and if u is σ
(
E
,
E
′
)-continuous on every bounded subset of E, then u is continuous on E with the initial topology of E. Thus, we obtain the following theorem.Theorem 15.
IfE is a DF space and a k-space, then we have
(54)
L
(
⨂
n
,
s
π
E
τ
0
;
F
)
≅
P
w
(
n
E
;
F
)
for every complete locally convex space F.IfE is a Banach space, then E is a DF-space and a k-space. Thus, we have the following corollary.Corollary 16.
IfE is a Banach space, then we have
(55)
L
(
⨂
n
,
s
π
E
τ
0
;
F
)
≅
P
w
(
n
E
;
F
)
for every complete locally convex space F.



---

*Source: 795016-2014-03-30.xml*