Abstract
Let and be locally convex spaces over and let be the space of all continuous -homogeneous polynomials from to . We denote by the -fold symmetric tensor product space of endowed with the projective topology. Then, it is well known that each polynomial is represented as an element in the space of all continuous linear mappings from to . A polynomial is said to be of weak type if, for every bounded set of , is weakly continuous on . We denote by the space of all -homogeneous polynomials of weak type from to . In this paper, in case that is a DF space, we will give the tensor product representation of the space .
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 and be complex vector spaces. Then the pair is called a dual pair if there exists a bilinear form: satisfying the following conditions: (1)If for every , .(2)If for every , .
We denote by (resp., ) the topology on (resp., ) defined by the subset of seminorms:
Let be a locally convex space. We denote by the set of all nontrivial continuous seminorms on . The topology on is called the weak topology of and the topology on is called the weak topology of . We denote by the family of all bounded subsets of . We denote by the seminorm on defined by for every . The strong topology on is the topology on defined by the set of seminorms on . We denote by the locally convex space endowed with the strong topology. We denote by the dual space of . Let be a subset of . We denote by the topological closure of the subset of for the topology . The polar set of is defined by
We denote by the bipolar set of for the dual pair . A subset of is said to be equicontinuous if there exists a neighborhood of in such that .
Lemma 1. Let be a bounded subset of a locally convex space . Then the following statements hold:(1) if is absolutely convex.(2) is compact with the topology .(3).(4)Let be an equicontinuous subset of for the dual pair . Then there exists an absolutely convex bounded subset of , such that .
Proof. Since and is -closed, . We shall show . We assume that . We denote by the locally convex space endowed with the topology . Since and is -closed absolutely convex, by Hahn-Banach theorem there exists such that
Thus, it is valid that and . Thus we have . Since , . Hence, we have .
We denote by the -closed absolutely convex hull of the set . By the statement we have
The polar set is an absolutely convex neighborhood of in . Therefore, is a -closed equicontinuous subset of . By Banach-Alaoglu theorem, is -compact. Since , is also -compact.
It is clear that . We shall show that . Let be a point of . Then, there exists an open neighborhood of in such that for every . By the definition of the space there exists an absolutely convex bounded subset such that . Thus, by the statement we have
Thus, we have .
Since is an equicontinuous subset of , there exists a neighborhood of in such that . Since is a neighborhood of in , there exists an absolutely convex bounded subset of such that . Thus, we have . This completes the proof.
A filter of a locally convex space is called a Cauchy filter if for every neighborhood of there exists an such that . A locally convex space is said to be complete if any Cauchy filter on converges to a point of . There exists the smallest complete locally convex space containing as a subspace. The locally convex space is called the completion of .
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 and be locally convex spaces and let be a positive integer. We denote by the space of all -linear mappings from the product space of -copies of into and denote by the space of all -linear mappings, which are -continuous on bounded subsets of , from the product space into . A mapping is called an -homogeneous polynomial from into if there exists an -linear mapping from into such that for every . If is an -homogeneous polynomial from into , there exists uniquely a symmetric -linear mapping . We denote by the space of all -homogeneous polynomials from into . We denote by (resp., ) the space of all continuous -homogeneous polynomials from (resp., all continuous -linear mappings from ) into . We denote by the space of all -continuous polynomials on each bounded subset of . We set
A polynomial belonging to is said to be of weak type.
Lemma 2. Let and be locally convex spaces and let be an -linear mapping belonging to . Let be absolutely convex bounded subsets of . Let be any point of -closure of for each with . We denote by the system of all -neighborhoods of in . Then, for any there exists such that for every .
Proof. We shall prove this lemma by induction on . Let . Then, the conclusion of this lemma is true since and is the induced topology of the topology onto .
We suppose that the conclusion of this lemma is true for all mappings belonging to . And we assume that for , the conclusion is not true. Then, there exists such that for every there is a point:
satisfying
Since is -continuous at on each absolutely convex bounded subset of and is the induced topology of onto , there exists such that
for every with . We choose a decreasing sequence of elements of by the process of steps as follows.
At the first step, we consider the -linear mapping:
belonging to . By the assumption of induction, there exists with such that
for every .
At the second step, we consider the -linear mapping
belonging to . By the assumption of induction, there exists with such that
for every point of
Repeating this process, at the th step, we consider the -linear mapping:
belonging to . By the assumption of induction, there exists with such that
for every . Then we have
where and .
If we set
we have
This is a contradiction by (13). This completes the proof.
Lemma 3. Let be a locally convex space, let be a complete locally convex space and let be an -linear mapping belonging to . Let be absolutely convex bounded subsets of . Then there exists a -continuous mapping from into such that on .
Proof. Let be any point of the -closure of for each with . At first, we shall show that a filter of is a Cauchy filter. Let be an arbitrary continuous seminorm of . By Lemma 2 there exists such that for every point of where and . Then, we have for every . Thus, the filter is a Cauchy filter. Since is complete and Hausdorff, there exists uniquely the limit point of the filter for every . We denote by the limit point of the filter for every . Then defines -continuous mapping from into with This completes the proof.
Lemma 4. Let be a locally convex space, let be a complete locally convex space, and let be an -linear mapping belonging to . Then, there exists with such that is -continuous on for all absolutely convex bounded subsets of .
Proof. By Lemma 3 for all absolutely convex bounded subsets of , there exists a -continuous mapping from to with on . If are absolutely convex bounded subsets of with for , we have on . Thus, by Lemma 1, we can define an n-linear mapping from into by setting on for all absolutely convex bounded subsets of . Then, the mapping 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 be a locally convex space and let be a complete locally convex space. Let . Then, the following statements are equivalent. (1).(2)For each absolutely convex bounded subset of , can be extended -continuously to , where is the bipolar set of for the dual pair .(3)There exists such that is -continuous on each equicontinuous subset of and .(4) is weakly uniformly continuous on every bounded subset of .
Proof. We shall show that implies . There exists a symmetric -linear mapping from into such that for every . By the polarization formula, we have . By Lemma 4 there exists with such that is -continuous on for all absolutely convex bounded subsets of . We define by for every . By Lemma 1, satisfies all required conditions of the statement .
implies since is equicontinuous for every absolutely convex bounded subset of .
We shall show that implies . Let be a bounded subset of . We denote by the absolutely convex hull of in . Then, is a bounded subset of , and . By statement , there exists a -continuous mapping from into such that on . Since by Lemma 1 is -compact, is uniformly -continuous on . Since on , is weakly uniformly continuous on . This implies . It is clear that implies . This completes the proof.
3. The Tensor Product Representation of Polynomials in Locally Convex Spaces
For any , we set for every , where is the permutation group of degree . Then, is a symmetric -linear mapping from to satisfying for every . We denote by the space of all symmetric -linear mappings from to . Let be the mapping from into defined by
For any , we define an -homogeneous polynomial by The mapping is surjective.
Theorem 6 (polarization formula). Let and let . If , then
By the above polarization formula, the mapping is a linear isomorphism.
We denote by the -fold tensor product space of . Let be the linear mapping from into defined by
for every . For any , there exists a unique such that the diagram
(37)
commutes. The mapping
is a linear isomorphism. Each element of has a representation of the form
However, this representation will never be unique. We denote by the mapping of into defined by for every .
Proposition 7. A mapping is an -homogeneous polynomial if and only if there exists such that the diagram
(41)
commutes.
For any , we set
We denote by the subspace of generated by . The space is called the -fold symmetric tensor product space of . Elements of are called -symmetric tensors. Clearly, every tensor of the form is a symmetric tensor. Moreover, each element in can be expressed as a finite sum (not necessarily unique) of the form
For any , there exists a unique such that the diagram
(44)
commutes. The mapping
is a linear isomorphism. Let , , and be, respectively, the spaces of continuous -homogeneous polynomials from into and the continuous symmetric -linear mappings from into . The restrictions
are well-defined. For each and , we set
is a seminorm on . We define the -topology or the projective topology on as the locally convex topology generated by . We denote by the space endowed with -topology and denote by the completion . Then, the following is valid (cf. Dineen [2]).
Proposition 8. Let be a locally convex space, then we have
Let be the family of all equicontinuous subsets of with respect to the dual pair . We denote by the family of subsets of defined by We denote by the topology on such that the family of all -open sets coincides with . By Theorem 5, the following is valid.
Proposition 9. A -homogeneous polynomial on is of weak type if and only if there exists a -continuous -homogeneous polynomial on such that on .
However, in general, the topology is not a locally convex topology (cf. Kōmura [7]).
Definition 10. A locally convex space 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 be a metrizable locally convex space. Then, the topology on is the topology of uniform convergence on all compact sets in .
Proof. The proof is on Köthe [10, 21–10].
If is a DF space, then is a Fréchet space. Therefore, by Proposition 11 the following is valid.
Proposition 12. The topology on is the topology of uniform convergence on all compact sets in .
We denote by the seminorm on defined by for every compact subsets of . We denote by the locally convex topology on defined by the set of seminorms: We denote by the locally convex space defined the set of seminorms: An -homogeneous polynomial on is -continuous on every equicontinuous subsets of if and only if is -continuous on . Thus, by Propositions 9 and 11, we have the following theorem.
Theorem 13. If an -homogeneous polynomial on is of weak type if and only if is -continuous in and continuous on with the initial topology of .
Then, we can obtain the following topological tensor product representation of polynomials of weak type.
Theorem 14. Let be a DF space, then we have for every complete locally convex space .
A topological space is called a -space if its topology is localized on its compact set;, that is, is open if and only if is open in , with the induced topology, for each compact subset of . A mapping from a -space into a topological space is continuous if and only if its restriction to each compact set is continuous.
Let be a mapping from a locally convex space into a locally convex space . If is a -space and if is -continuous on every bounded subset of , then is continuous on with the initial topology of . Thus, we obtain the following theorem.
Theorem 15. If is a DF space and a -space, then we have for every complete locally convex space .
If is a Banach space, then is a DF-space and a -space. Thus, we have the following corollary.
Corollary 16. If is a Banach space, then we have for every complete locally convex space .
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).