Let E and F be locally convex spaces over C and let P(nE;F) be the space of all continuous n-homogeneous polynomials from E to F. We denote by ⨂n,s,πE the n-fold symmetric tensor product space of E endowed with the projective topology. Then, it is well known that each polynomial p∈P(nE;F) is represented as an element in the space L(⨂n,s,πE;F) of all continuous linear mappings from ⨂n,s,πE to F. A polynomial p∈P(nE;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 Pw(nE;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 Pw(nE;F).

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.

Let E1 and E2 be complex vector spaces. Then the pair 〈E1,E2〉 is called a dual pair if there exists a bilinear form:
(1)(x1,x2)⟶〈x1,x2〉((x1,x2)∈E1×E2)
satisfying the following conditions:

If 〈x1,x2〉=0 for every x2∈E2, x1=0.

If 〈x1,x2〉=0 for every x1∈E1, x2=0.

We denote by σ(E1,E2) (resp., σ(E2,E1)) the topology on E1 (resp., E2) defined by the subset of seminorms:
(2){|〈·,x2〉|;x2∈E2}(resp.{|〈x1,·〉|;x1∈E1}).

Let E be a locally convex space. We denote by cs(E) the set of all nontrivial continuous seminorms on E. The topology σ(E,E′) on E is called the weak topology of E and the topology σ(E′,E) on E′ is called the weak * 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). The strong topology on E′ is the topology on E′ defined by the set of seminorms {∥∥B;B∈ℬ(E)} on E′. We denote by Eβ′ the locally convex space E′ endowed with the strong topology. We denote by E′′ the dual space of Eβ′. 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′). The polar set A∘ of A is defined by
(4)A∘={x′∈E′;supx∈A|〈x,x′〉|≤1}.

We denote by A∘∘ the bipolar set of A for the dual pair 〈E′′,E′〉. A subset S of E′ is said to be equicontinuous if there exists a neighborhood V of 0 in E such that S⊂V∘.

Lemma 1.

Let M be a bounded subset of a locally convex space E. Then the following statements hold:

M¯=M∘∘ if M is absolutely convex.

M¯ is compact with the topology σ(E′′,E′).

E′′=⋃M∈ℬ(E)M¯.

Let A 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 x0′′∉M¯. We denote by Eσ(E′′,E′)′′ the locally convex space E′′ endowed with the topology σ(E′′,E′). Since (Eσ(E′′,E′)′′)′=E′ and M¯ is σ(E′′,E′)-closed absolutely convex, by Hahn-Banach theorem there exists x′∈E′ such that
(5)supx′′∈M¯|〈x′′,x′〉|≤1,|〈x0′′,x′〉|>1.
Thus, it is valid that x′∈M¯∘ and x0′′∉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 Eβ′. 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 Eβ′ such that |〈x,x′′〉|≤1 for every x∈V. By the definition of the space Eβ′ 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 Eβ′ such that A⊂V∘. Since V is a neighborhood of 0 in Eβ′, 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 filter ℱ={Fα} of a locally convex space E is called a Cauchy 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 be complete 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 the completion 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 La(nE;F) the space of all n-linear mappings from the product space En of n-copies of E into F and denote by Law(nE;F) the space of all n-linear mappings, which are σ(E,E′)-continuous on bounded subsets of En, from the product space En 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 Pa(nE;F) the space of all n-homogeneous polynomials from E into F. We denote by P(nE;F) (resp., L(nE;F)) the space of all continuous n-homogeneous polynomials from E (resp., all continuous n-linear mappings from En) into F. We denote by Paw(nE;F) the space of all σ(E,E′)-continuous polynomials on each bounded subset of E. We set
(9)Pw(nE;F)=P(nE;F)∩Paw(nE;F),Lw(nE;F)=L(nE;F)∩Law(nE;F).

A polynomial belonging to Pw(nE;F) is said to be of weak type.

Lemma 2.

Let E and F be locally convex spaces and let u be an n-linear mapping belonging to Lw(nE;F). Let A1,…,An be absolutely convex bounded subsets of E. Let ai be any point of σ(E′′,E′)-closure A¯i of Ai for each i with 2≤i≤n. We denote by 𝒩w*(E′′)(0) the system of all σ(E′′,E′)-neighborhoods of 0 in E′′. Then, for any α∈cs(F) there exists V∈𝒩w*(E′′)(0) such that
(10)α(u(x1,…,xn))<1,
for every (x1,…,xn)∈(V∩A1)×((a2+V)∩A2)×⋯×((an+V)∩An).

Proof.

We shall prove this lemma by induction on n. Let n=1. Then, the conclusion of this lemma is true since 0∈A1 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 to Lw(n-1E;F). And we assume that for u∈Lw(nE;F), the conclusion is not true. Then, there exists α∈cs(F) such that for every V∈𝒩w*(E′′)(0) there is a point:
(11)(x1V,x2V,…,xnV)∈(V∩A1)×((a2+V)∩A2)×⋯×((an+V)∩An),
satisfying
(12)α(u(x1V,x2V,…,xnV))≥1.

Since u is σ(E,E′)-continuous at (0,0,…,0) on each absolutely convex bounded subset of En and σ(E,E′) is the induced topology of σ(E′′,E′) onto E, there exists W∈𝒩w*(E′′)(0) such that
(13)α(u(x1,x2,…,xn))≤14
for every xi∈2(W∩Ai) with 1≤i≤n. We choose a decreasing sequence W⊃W(1)⊃W(2)⊃⋯⊃W(n-1) of elements of 𝒩w*(E′′)(0) by the process of (n-1) steps as follows.

At the first step, we consider the (n-1)-linear mapping:
(14)(z1,…,zn-1)⟶u(z1,…,zn-1,xnW),
belonging to Lw(n-1E;F). By the assumption of induction, there exists W(1)∈𝒩w*(E′′)(0) with W(1)⊂W such that
(15)α(u(z1,…,zn-1,xnW))≤12n
for every (z1,…,zn-1)∈(W(1)∩A1)×((a2+W(1))∩A2)×⋯×((an-1+W(1))∩An-1).

At the second step, we consider the (n-1)-linear mapping
(16)(z1,…,zn-2,zn)⟶u(z1,…,zn-2,xn-1W(1),zn)
belonging to Lw(n-1E;F). By the assumption of induction, there exists W(2)∈𝒩w*(E′′)(0) with W(2)⊂W(1) such that
(17)α(u(z1,…,zn-2,xn-1W(1),zn))≤12n
for every point (z1,…,zn-2,zn) of
(18)(W(2)∩A1)×((a2+W(2))∩A2)×⋯×((an-2+W(2))∩An-2)×((an+W(2))∩An).

Repeating this process, at the (n-1)th step, we consider the (n-1)-linear mapping:
(19)(z1,z3,…,zn)⟶u(z1,x2W(n-2),z3,…,zn),
belonging to Lw(n-1E;F). By the assumption of induction, there exists W(n-1)∈𝒩w*(E′′)(0) with W(n-1)⊂W(n-2) such that
(20)α(u(z1,x2W(n-2),z3,…,zn))≤12n
for every (z1,z3…,zn)∈(W(n-1)∩A1)×((a3+W(n-1))∩A3)×⋯×((an+W(n-1))∩An). Then we have
(21)α(u(x1W(n-1),x2W(n-1)-x2W(n-2),…,xn-1W(n-1)mmm-xn-1W(1),xnW(n-1)-xnW))≥α(u(x1W(n-1),x2W(n-1),…,xnW(n-1)))-∑(ki)∈Kα(u(x1W(n-1),x2W(k2),…,xnW(kn)))≥1-(2n-1-1)×12n≥12,
where K={(k2,…,kn);ki∈{n-1,n-i},i=2,…,n}∖{(n-1,…,n-1)} and W(0)=W.

If we set
(22)(y1,y2,…,yn)=(x1W(n-1),x2W(n-1)-x2W(n-2),…,xnW(n-1)-xnW),
we have
(23)α(u(y1,…,yn))≥12,(y1,y2,…,yn)∈(W∩A1)×{2(W∩A2)}×⋯×{2(W∩An)}.

This is a contradiction by (13). This completes the proof.

Lemma 3.

Let E be a locally convex space, let F be a complete locally convex space and let u be an n-linear mapping belonging to Lw(nE;F). Let A1,…,An 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 A1×⋯×An.

Proof.

Let ai be any point of the σ(E′′,E′)-closure A¯i of Ai for each i with 1≤i≤n. At first, we shall show that a filter of F(24)ℱ(a1,…,an)={V∈𝒩w*(E′′)u((a1+V)∩A1,…,(an+V)∩An);V∈𝒩w*(E′′)(0)}
is a Cauchy filter. Let α be an arbitrary continuous seminorm of F. By Lemma 2 there exists V∈𝒩w*(E′′)(0) such that
(25)α(u(x1,x2,…,xn))<1n,
for every point (x1,x2,…,xn) of
(26)⋃i=1n(∏j=1n((γijaj+2V)∩2Aj)),
where γij=1(i≠j) and γij=0(i=j). Then, we have
(27)α(u(x1,x2,…,xn)-u(y1,y2,…,yn))≤α(u(x1-y1,x2,…,xn))+α(u(y1,x2-y2,x3,…,xn))+α(u(y1,y2,x3-y3,x4,…,xn))+⋯+α(u(y1,y2,…,yn-1,xn-yn))≤n×1n=1,
for every (x1,…,xn),(y1,…,yn)∈∏i=1n((ai+V)∩Ai). Thus, the filter ℱ(a1,…,an) is a Cauchy filter. Since F is complete and Hausdorff, there exists uniquely the limit point of the filter ℱ(a1,…,an) for every (a1,…,an)∈A¯1×⋯×A¯n. We denote by u~A¯1×⋯×A¯n(a1,…,an) the limit point of the filter ℱ(a1,…,an) for every (a1,…,an)∈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=uonA1×⋯×An.
This completes the proof.

Lemma 4.

Let E be a locally convex space, let F be a complete locally convex space, and let u be an n-linear mapping belonging to Lw(nE;F). Then, there exists u~∈La(nE′′;F) with u~∣En=u such that u~ is σ(E′′,E′)-continuous on A¯1×⋯×A¯n for all absolutely convex bounded subsets A1,…,An of E.

Proof.

By Lemma 3 for all absolutely convex bounded subsets A1,…,An 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 A1×⋯×An. If A1,…,An,B1,…,Bn 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 an n-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 A1,…,An 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.

Let E be a locally convex space and let F be a complete locally convex space. Let p∈P(nE;F). Then, the following statements are equivalent.

p∈Pw(nE;F).

For each absolutely convex bounded subset M 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′〉.

There exists p~∈Pa(nE′′;F) such that p~ is σ(E′′,E′)-continuous on each equicontinuous subset of E′′ and p~∣E=p.

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 En into F such that p(x)=u(x,…,x) for every x∈E. By the polarization formula, we have u∈Lw(nE,F). By Lemma 4 there exists u~∈La(nE′′;F) with u~∣En=u such that u~ is σ(E′′,E′)-continuous on A¯1×⋯×A¯n for all absolutely convex bounded subsets A1,…,An of E. We define p~∈Pa(nE′′;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 1M¯ 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 any u∈La(nE;F), we set
(30)s(u)(x1,…,xn)=1n!∑σ∈Snu(xσ(1),…,xσ(n))
for every (x1,…,xn)∈En, where Sn is the permutation group of degree n. Then, s(u) is a symmetric n-linear mapping from En to F satisfying
(31)u(x,…,x)=s(u)(x,…,x)
for every x∈E. We denote Las(nE;F) by the space of all symmetric n-linear mappings from En to F. Let Δn be the mapping from E into En defined by
(32)Δn(x)=(x,…,x)foreveryx∈E.

For any u∈La(nE;F), we define an n-homogeneous polynomial Δn*(u) by
(33)Δn*(u)=u∘Δn.
The mapping
(34)Δn*:Las(nE;F)⟶Pa(nE;F)
is surjective.

Theorem 6 (polarization formula).

Let p∈Pa(nE;F) and let u∈Las(nE;F). If Δn*(u)=p, then
(35)u(x1,…,xn)=12nn!∑ϵi=±1ϵ1⋯ϵnp(∑i=1nϵixi).

By the above polarization formula, the mapping Δn*:Las(nE;F)→Pa(nE;F) is a linear isomorphism.

We denote by ⨂nE the n-fold tensor product space of E. Let in be the linear mapping from En into ⨂nE defined by
(36)in(x1,…,xn)=x1⊗⋯⊗xn
for every (x1,…,xn)∈En. For any u∈La(nE;F), there exists a unique in*(u)∈La(⨂nE;F) such that the diagram
(37)
commutes. The mapping
(38)in*:La(nE;F)⟶La(⨂nE;F)
is a linear isomorphism. Each element of ⨂nE has a representation of the form
(39)∑i=1ℓxi,1⊗xi,2⊗⋯⊗xi,n.

However, this representation will never be unique. We denote by δn the mapping of E into ⨂nE defined by
(40)δn(x)=x⊗⋯⊗x
for every x∈E.

Proposition 7.

A mapping p:E→F is an n-homogeneous polynomial if and only if there exists T∈La(nE;F) such that the diagram
(41)
commutes.

For any x1⊗⋯⊗xn∈⨂nE, we set
(42)s(x1⊗⋯⊗xn)=1n!∑σ∈Snxσ(1)⊗⋯⊗xσ(n).

We denote by ⨂n,sE the subspace of ⨂nE generated by s(x1⊗⋯⊗xn),xi∈E. The space ⨂n,sE is called the n-fold symmetric tensor product space of E. Elements of ⨂n,sE are called n-symmetric tensors. Clearly, every tensor of the form x⊗⋯⊗x is a symmetric tensor. Moreover, each element θ in ⨂n,sE can be expressed as a finite sum (not necessarily unique) of the form
(43)∑ixi⊗⋯⊗xi.
For any p∈Pa(nE;F), there exists a unique jn*(p)∈La(⨂n,sE;F) such that the diagram
(44)
commutes. The mapping
(45)jn*:Pa(nE;F)⟶La(⨂n,sE;F)
is a linear isomorphism. Let P(nE;F), L(nE;F), and Ls(nE;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)Δn*:L(nE;F)⟶P(nE;F),Δn*:Ls(nE;F)⟶P(nE;F),s:L(nE;F)⟶Ls(nE;F)
are well-defined. For each α∈cs(E) and θ=∑ixi⊗⋯⊗xi∈⨂n,sE, we set
(47)πα,n(θ)=inf{∑iα(xi)n∣θ=∑ixi⊗⋯⊗xi}.πα,n is a seminorm on ⨂n,sE. We define the π-topology or the projective topology on ⨂n,sE as the locally convex topology generated by {πα,n}α∈cs(E). We denote by ⨂n,s,πE the space endowed with π-topology and denote by ⨂¯n,s,πE the completion ⨂n,s,πE. Then, the following is valid (cf. Dineen [2]).

Proposition 8.

Let E be a locally convex space, then we have
(48)L(nE;F)≅L(⨂n.πE;F),Ls(nE;F)≅L(⨂n.s,πE;F)≅P(nE;F).

Let ℰ(E′′) be the family of all equicontinuous subsets of E′′ with respect to the dual pair 〈E′′,E′〉. We denote by 𝒪w*,ϵ the family of subsets of E′′ defined by
(49)𝒪w*,ϵ={V∣V⊂E′′,V∩Mareσ(E′′,E′)-openinMforallM∈ℰ(E′′)}.
We denote by τw*,ϵ the topology on E′′ such that the family of all τw*,ϵ-open sets coincides with 𝒪w*,ϵ. By Theorem 5, the following is valid.

Proposition 9.

A n-homogeneous polynomial p on E is of weak type if and only if there exists a τw*,ϵ-continuous n-homogeneous polynomial p~ on E′′ such that p~=p on E.

However, in general, the topology τw*,β is not a locally convex topology (cf. Kōmura [7]).

Definition 10.

A locally convex space E 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).

Let E be a metrizable locally convex space. Then, the topology τw*,ϵ on E′ is the topology of uniform convergence on all compact sets in E.

Proof.

The proof is on Köthe [10, § 21–10].

If E is a DF space, then Eβ′ is a Fréchet space. Therefore, by Proposition 11 the following is valid.

Proposition 12.

The topology τw*,ϵ on E′′ is the topology of uniform convergence on all compact sets in E′.

We denote by αK the seminorm on E′′ defined by
(50)αK(x′′)=sup{|〈x′′,x′〉|∣x′∈K}
for every compact subsets K of Eβ′. We denote by τ0 the locally convex topology on E′′ defined by the set of seminorms:
(51){αK∣KiseverycompactsetofE′}.
We denote by Eτ0 the locally convex space E defined the set of seminorms:
(52){αK|E∣KiseverycompactsetofE′}.
An n-homogeneous polynomial p on E′′ is σ(E′′,E′)-continuous on every equicontinuous subsets of E′′ if and only if p is τw*,ϵ-continuous on E′′. Thus, by Propositions 9 and 11, we have the following theorem.

Theorem 13.

If an n-homogeneous polynomial p on E is of weak type if and only if p is τo-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.

Let E be a DF space, then we have
(53)L(⨂n,sπEτ0;F)∩L(⨂n,sπE;F)≅Pw(nE;F)
for every complete locally convex space F.

A topological space X 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.

Let u 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.

If E is a DF space and a k-space, then we have
(54)L(⨂n,sπEτ0;F)≅Pw(nE;F)
for every complete locally convex space F.

If E is a Banach space, then E is a DF-space and a k-space. Thus, we have the following corollary.

Corollary 16.

If E is a Banach space, then we have
(55)L(⨂n,sπEτ0;F)≅Pw(nE;F)
for every complete locally convex space F.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Kwang Ho Shon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT and Future Planning (2013R1A1A2008978).

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