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

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The Tensor Product Representation of Polynomials of Weak Type in a DF-Space Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 795016, 7 pages http://dx.doi.org/10.1155/2014/795016 Research Article The Tensor Product Representation of Polynomials of Weak Type in a DF-Space Masaru Nishihara1 and Kwang Ho Shon2 1 Department of Computer Science and Engineering, Faculty of Information Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan 2 Department of Mathematics, College of Natural Sciences, Pusan National University, Busan 609-735, Republic of Korea Correspondence should be addressed to Kwang Ho Shon; [email protected] Received 15 December 2013; Accepted 19 February 2014; Published 27 March 2014 Academic Editor: Zong-Xuan Chen Copyright © 2014 M. Nishihara and K. H. Shon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let and be locally convex spaces over C and let ( ; ) be the space of all continuous -homogeneous polynomials from to .Wedenoteby⨂,, 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 .Wedenoteby( ; ) 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 by B() thefamilyofallboundedsubsetsof. We denote by ‖‖ the seminorm on defined by In this section, we collect some notations, some definitions, = {⟨, ⟩ ;∈}, and some basic properties of locally convex spaces which we sup (3) use throughout this paper. Let 1 and 2 be complex vector spaces. Then the pair for every ∈B().Thestrong topology on is the topology ⟨1,2⟩ is called a dual pair if there exists a bilinear form: on defined by the set of seminorms {‖ ‖; ∈ B()} on . Wedenote by the locally convex space endowed with (1,2) → ⟨1,2⟩((1,2) ∈1 ×2) (1) the strong topology. We denote by the dual space of .Let be a subset of . We denote by the topological closure of satisfying the following conditions: the subset of ⊂ for the topology ( , ).Thepolar ∘ (1) If ⟨1,2⟩=0for every 2 ∈2, 1 =0. set of is defined by (2) If ⟨1,2⟩=0for every 1 ∈1, 2 =0. ∘ ={ ∈; sup ⟨, ⟩ ≤1}. (4) ∈ We denote by (1,2) (resp., (2,1))thetopologyon1 ∘∘ (resp., 2) defined by the subset of seminorms: We denote by the bipolar set of for the dual pair ⟨ , ⟩.Asubset of is said to be equicontinuous if there {⟨⋅, 2⟩ ;2 ∈2}(resp.{⟨1,⋅⟩ ;1 ∈1}) . ∘ (2) exists a neighborhood of 0 in such that ⊂. () Let be a locally convex space. We denote by the set Lemma 1. Let beaboundedsubsetofalocallyconvexspace of all nontrivial continuous seminorms on .Thetopology . Then the following statements hold: (, ) on is called the weak topology of and the topology ∗ ∘∘ ( ,)on is called the weak topology of . We denote (1) = if is absolutely convex. 2 Abstract and Applied Analysis (2) is compact with the topology ( , ). 2. The Extension of Polynomial Mappings of Weak Type (3) = ⋃∈B() . In this section, we will give basic properties of polynomial (4) Let be an equicontinuous subset of for the dual mappings on locally convex spaces and discuss the extension pair ⟨ , ⟩. Then there exists an absolutely convex ∘∘ of weak type on locally convex spaces. For more detailed bounded subset of ,suchthat⊂ . properties of polynomials on locally convex spaces, see (1) ⊂∘∘ ∘∘ (,) ⊂ Dineen [1, 2]andMujica[3]. Let and be locally convex Proof. Since and is -closed, (; ) ∘∘ ∘∘ ⊂ ∉ spaces and let be a positive integer. We denote by .Weshallshow . We assume that 0 . the space of all -linear mappings from the product space We denote by ( , ) the locally convex space endowed of -copies of into and denote by ( ; ) the space with the topology ( , ).Since((,)) =and of all -linear mappings, which are (, )-continuous on is ( , )-closed absolutely convex, by Hahn-Banach bounded subsets of ,fromtheproductspace into .A : → theorem there exists ∈ such that mapping is called an -homogeneous polynomial from into if there exists an -linear mapping from into such that sup ⟨ , ⟩ ≤1, ∈ (5) () =(,...,) , (8) ⟨,⟩ >1. 0 for every ∈.If is an -homogeneous polynomial from into , there exists uniquely a symmetric -linear ∘ ∘∘ (; ) Thus, it is valid that ∈ and 0 ∉ .Thuswehave mapping . We denote by the space of all - ∘∘ ∘∘ ∘∘ ∘∘ homogeneous polynomials from into . We denote by ⊂ .Since ⊂ , ⊂ .Hence,wehave= ∘∘ ( ; ) (resp., ( ; )thespaceofallcontinuous) - . homogeneous polynomials from (resp., all continuous - (2) We denote by Γ() the ( , )-closed absolutely linear mappings from )into. We denote by ( ; ) convex hull of the set . By the statement (1) we have the space of all (, )-continuous polynomials on each Γ () =∘∘. bounded subset of .Weset (6) ( ; ) =( ; ) ∩ ( ; ) , ∘ (9) The polar set is an absolutely convex neighborhood of 0 ( ; ) =( ; ) ∩ ( ; ) . in . Therefore, Γ() is a ( , )-closed equicontinuous subset of . By Banach-Alaoglu theorem, Γ() is ( , )- A polynomial belonging to ( ; ) is said to be of weak compact. Since ⊂Γ(), is also ( , )-compact. type. (3) It is clear that ⋃∈B() ⊂ .Weshallshowthat Lemma 2. Let and be locally convex spaces and let be ⊂⋃ .Let be a point of . Then, there exists ∈B() an -linear mapping belonging to ( ; .Let) 1,..., be 0 |⟨, ⟩| ≤ 1 an open neighborhood of in such that absolutely convex bounded subsets of .Let be any point of for every ∈. By the definition of the space there exists ( , )-closure of for each with 2≤≤.Wedenote an absolutely convex bounded subset ∈B() such that by N∗ ( )(0) the system of all ( , )-neighborhoods of 0 ∘ ⊂. Thus, by the statement (1) we have in .Then,forany ∈ () there exists ∈N∗ ( )(0) such that ∘ ∘∘ ∈ ⊂ = . (7) ((1,...,)) < 1, (10) for every (1,...,) ∈ (∩1)×((2 +)∩2)×⋅⋅⋅×(( + Thus, we have =⋃∈B() . ) ∩ ). (4) Since is an equicontinuous subset of , there exists 0 ⊂∘ a neighborhood of in such that .Since is a Proof. Weshallprovethislemmabyinductionon.Let= 0 neighborhood of in , there exists an absolutely convex 1. Then, the conclusion of this lemma is true since 0∈1 ∘ bounded subset of such that ⊂.Thus,wehave and (, ) is the induced topology of the topology ( , ) ∘ ∘∘ ⊂ ⊂ .Thiscompletestheproof. onto . We suppose that the conclusion of this lemma is true for −1 all mappings belonging to ( ; .) And we assume that AfilterF ={} of a locally convex space is called a for ∈( ; ,) the conclusion is not true. Then, there Cauchy filter if for every neighborhood of 0 there exists an exists ∈ () such that for every ∈N∗ ( )(0) there ∈F such that {−;,∈}⊂. A locally convex space is a point: is said to be complete if any Cauchy filter on converges to a point of . There exists the smallest complete locally convex (1,2,...,)∈(∩1)×((2 +)∩2) space ̃ containing as a subspace. The locally convex space (11) ×⋅⋅⋅×(( +)∩ ), ̃ is called the completion of . Abstract and Applied Analysis 3 satisfying for every (1,3 ...,) ∈ ((−1)∩1)×((3 +(−1))∩ 3) × ⋅ ⋅ ⋅ × (( +(−1))∩).Thenwehave ( (1,2,...,)) ≥1. (12) ((1(−1),2(−1) −2(−2),...,−1(−1) Since is (, )-continuous at (0,0,...,0) on each − , − )) −1(1) (−1) absolutely convex bounded subset of and (, ) is the induced topology of ( , ) onto , there exists ∈ ≥((1(−1),2(−1),...,(−1))) N∗ ( )(0) such that (21) − ∑ (( , ,..., )) 1(−1) 2(2) () 1 ( )∈ ( ( , ,..., )) ≤ 1 2 4 (13) 1 1 ≥1−(2−1 −1)× ≥ , 2 2 for every ∈2(∩) with 1≤≤.Wechoosea decreasing sequence ⊃ (1) ⊃ (2) ⊃ ⋅ ⋅ ⋅ ⊃ (−1) where ={(2,...,); ∈{−1,−},=2,...,}\{(− of elements of N∗( )(0) by the process of ( − 1) steps as 1,...,−1)}and (0) =. follows. If we set At the first step, we consider the ( − 1)-linear mapping: (1,2,...,)= (1(−1),2(−1) −2(−2),..., (1,...,−1) → (1,...,−1,) , (14) (22) (−1) −), −1 belonging to ( ; .) By the assumption of induction, we have there exists (1) ∈ N∗ ( )(0) with (1) ⊂ such that 1 (( ,..., )) ≥ , 1 1 (( ,..., , )) ≤ 2 1 −1 (15) 2 (23) (1,2,...,)∈ (∩1)×{2(∩2)} for every (1,...,−1) ∈ ((1) ∩ 1) × ((2 + (1)) ∩ 2)× ×⋅⋅⋅×{2(∩)} . ⋅ ⋅ ⋅ × ((−1 + (1)) ∩ −1). (−1) Atthesecondstep,weconsiderthe -linear mapping This is a contradiction by (13). This completes the proof. (1,...,−2,) → (1,...,−2,−1(1),) (16) Lemma 3. Let be a locally convex space, let be a complete −1 belonging to ( ; .) By the assumption of induction, locally convex space and let be an -linear mapping belonging (; ) ,..., there exists (2) ∈ N∗ ( )(0) with (2) ⊂ (1) such to .Let 1 be absolutely convex bounded (,) that subsets of . Then there exists a -continuous mapping ̃ ×⋅⋅⋅× from 1 ×⋅⋅⋅× into such that ̃ ×⋅⋅⋅× =on 1 1 ×⋅⋅⋅× 1 (( ,..., , , )) ≤ 1 . 1 −2 −1(1) 2 (17) Proof. Let be any point of the ( , )-closure of for for every point (1,...,−2,) of each with 1≤≤. At first, we shall show that a filter of F ( ,..., )= {(( +)∩ ,...,( +)∩ ); ( (2) ∩1)×((2 +(2))∩2) 1 1 1 ×⋅⋅⋅×((−2 +(2))∩−2) × (( +(2))∩). ∈N∗ ( ) (0)} (18) (24) Repeating this process, at the ( − 1)th step, we consider ( − 1) is a Cauchy filter. Let be an arbitrary continuous seminorm the -linear mapping: of .ByLemma 2 there exists ∈N∗ ( )(0)
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