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.

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

1. Notations and Preliminaries by B(𝐸) thefamilyofallboundedsubsetsof𝐸. We denote by 󸀠 ‖‖𝐵 the seminorm on 𝐸 defined by In this section, we collect some notations, some definitions, 󵄩 󵄩 󵄨 󵄨 󵄩𝑥󸀠󵄩 = {󵄨⟨𝑥, 𝑥󸀠⟩󵄨 ;𝑥∈𝐵}, and some basic properties of locally convex spaces which we 󵄩 󵄩𝐵 sup 󵄨 󵄨 (3) use throughout this paper. 󸀠 Let 𝐸1 and 𝐸2 be complex vector spaces. Then the pair for every 𝐵∈B(𝐸).Thestrong topology on 𝐸 is the topology 󸀠 ⟨𝐸1,𝐸2⟩ is called a dual pair if there exists a bilinear form: on 𝐸 defined by the set of seminorms {‖ ‖𝐵;𝐵 ∈ B(𝐸)} on 󸀠 󸀠 󸀠 𝐸 . Wedenote by 𝐸𝛽 the locally convex space 𝐸 endowed with (𝑥1,𝑥2) 󳨀→ ⟨𝑥1,𝑥2⟩((𝑥1,𝑥2) ∈𝐸1 ×𝐸2) 󸀠󸀠 󸀠 (1) 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 satisfying the following conditions: 󸀠󸀠 󸀠󸀠 󸀠 the subset 𝐴 of 𝐸⊂𝐸 for the topology 𝜎(𝐸 ,𝐸 ).Thepolar 𝐴∘ 𝐴 (1) If ⟨𝑥1,𝑥2⟩=0for every 𝑥2 ∈𝐸2, 𝑥1 =0. set of is defined by

(2) If ⟨𝑥1,𝑥2⟩=0for every 𝑥1 ∈𝐸1, 𝑥2 =0. 󵄨 󵄨 ∘ 󸀠 󸀠 󵄨 󸀠 󵄨 𝐴 ={𝑥 ∈𝐸; sup 󵄨⟨𝑥, 𝑥 ⟩󵄨 ≤1}. (4) 𝑥∈𝐴 We denote by 𝜎(𝐸1,𝐸2) (resp., 𝜎(𝐸2,𝐸1))thetopologyon𝐸1 𝐸 ∘∘ (resp., 2) defined by the subset of seminorms: We denote by 𝐴 the bipolar set of 𝐴 for the dual pair 󸀠󸀠 󸀠 󸀠 󵄨 󵄨 󵄨 󵄨 ⟨𝐸 ,𝐸 ⟩.Asubset𝑆 of 𝐸 is said to be equicontinuous if there {󵄨⟨⋅, 𝑥2⟩󵄨 ;𝑥2 ∈𝐸2}(resp.{󵄨⟨𝑥1,⋅⟩󵄨 ;𝑥1 ∈𝐸1}) . ∘ 󵄨 󵄨 󵄨 󵄨 (2) exists a neighborhood 𝑉 of 0 in 𝐸 such that 𝑆⊂𝑉. 𝐸 𝑐𝑠(𝐸) Let be a locally convex space. We denote by the set Lemma 1. Let 𝑀 beaboundedsubsetofalocallyconvexspace of all nontrivial continuous seminorms on 𝐸.Thetopology 󸀠 𝐸. Then the following statements hold: 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 (1) 𝑀=𝑀 if 𝑀 is absolutely convex. 2 Abstract and Applied Analysis

󸀠󸀠 󸀠 (2) 𝑀 is compact with the topology 𝜎(𝐸 ,𝐸 ). 2. The Extension of Polynomial Mappings of 󸀠󸀠 Weak Type (3) 𝐸 = ⋃𝑀∈B(𝐸) 𝑀. 󸀠󸀠 In this section, we will give basic properties of polynomial (4) Let 𝐴 be an equicontinuous subset of 𝐸 for the dual 󸀠󸀠 󸀠 mappings on locally convex spaces and discuss the extension pair ⟨𝐸 ,𝐸 ⟩. Then there exists an absolutely convex ∘∘ of weak type on locally convex spaces. For more detailed bounded subset 𝑀 of 𝐸,suchthat𝐴⊂𝑀 . properties of polynomials on locally convex spaces, see 𝐸 𝐹 (1) 𝑀⊂𝑀∘∘ 𝑀∘∘ 𝜎(𝐸󸀠󸀠,𝐸󸀠) 𝑀⊂ Dineen [1, 2]andMujica[3]. Let and be locally convex Proof. Since and is -closed, 𝑛 𝐿 (𝑛𝐸; 𝐹) 𝑀∘∘ 𝑀∘∘ ⊂ 𝑀 𝑥󸀠󸀠 ∉ 𝑀 spaces and let be a positive integer. We denote by 𝑎 .Weshallshow . We assume that 0 . 𝑛 𝐸𝑛 󸀠󸀠 󸀠󸀠 the space of all -linear mappings from the product space 𝐸 󸀠󸀠 󸀠 𝐸 𝑛 We denote by 𝜎(𝐸 ,𝐸 ) the locally convex space endowed of 𝑛-copies of 𝐸 into 𝐹 and denote by 𝐿𝑎𝑤( 𝐸; 𝐹) the space 󸀠󸀠 󸀠 󸀠󸀠 󸀠 󸀠 󸀠 with the topology 𝜎(𝐸 ,𝐸 ).Since(𝐸𝜎(𝐸󸀠󸀠,𝐸󸀠)) =𝐸and of all 𝑛-linear mappings, which are 𝜎(𝐸, 𝐸 )-continuous on 󸀠󸀠 󸀠 𝐸𝑛 𝐸𝑛 𝐹 𝑀 is 𝜎(𝐸 ,𝐸 )-closed absolutely convex, by Hahn-Banach bounded subsets of ,fromtheproductspace into .A 󸀠 󸀠 𝑝:𝐸 →𝐹 𝑛 theorem there exists 𝑥 ∈𝐸 such that mapping is called an -homogeneous polynomial from 𝐸 into 𝐹 if there exists an 𝑛-linear mapping 𝑢 from 𝐸 󵄨 󵄨 󵄨 󸀠󸀠 󸀠 󵄨 into 𝐹 such that sup 󵄨⟨𝑥 ,𝑥 ⟩󵄨 ≤1, 𝑥󸀠󸀠∈𝑀 (5) 𝑃 (𝑥) =𝑢(𝑥,...,𝑥) , (8) 󵄨 󵄨 󵄨⟨𝑥󸀠󸀠,𝑥󸀠⟩󵄨 >1. 󵄨 0 󵄨 for every 𝑥∈𝐸.If𝑝 is an 𝑛-homogeneous polynomial from 𝐸 into 𝐹, there exists uniquely a symmetric 𝑛-linear 󸀠 ∘ 󸀠󸀠 ∘∘ 𝑢 𝑃 (𝑛𝐸; 𝐹) 𝑛 Thus, it is valid that 𝑥 ∈ 𝑀 and 𝑥0 ∉ 𝑀 .Thuswehave mapping . We denote by 𝑎 the space of all - ∘∘ ∘∘ ∘∘ ∘∘ homogeneous polynomials from 𝐸 into 𝐹. We denote by 𝑀 ⊂ 𝑀.Since𝑀 ⊂ 𝑀 , 𝑀 ⊂ 𝑀.Hence,wehave𝑀= 𝑛 𝑛 ∘∘ 𝑃( 𝐸; 𝐹) (resp., 𝐿( 𝐸; )thespaceofallcontinuous𝐹) 𝑛- 𝑀 . 󸀠󸀠 󸀠 homogeneous polynomials from 𝐸 (resp., all continuous 𝑛- (2) We denote by Γ(𝑀) the 𝜎(𝐸 ,𝐸 )-closed absolutely 𝑛 𝑛 linear mappings from 𝐸 )into𝐹. We denote by 𝑃𝑎𝑤( 𝐸; 𝐹) convex hull of the set 𝑀. By the statement (1) we have 󸀠 the space of all 𝜎(𝐸, 𝐸 )-continuous polynomials on each 𝐸 Γ (𝑀) =𝑀∘∘. bounded subset of .Weset (6) 𝑛 𝑛 𝑛 𝑃𝑤 ( 𝐸; 𝐹) =𝑃( 𝐸; 𝐹)𝑎𝑤 ∩𝑃 ( 𝐸; 𝐹) , ∘ (9) The polar set 𝑀 is an absolutely convex neighborhood of 0 𝑛 𝑛 𝑛 󸀠 󸀠󸀠 󸀠 𝐿𝑤 ( 𝐸; 𝐹) =𝐿( 𝐸; 𝐹) ∩𝐿𝑎𝑤 ( 𝐸; 𝐹) . in 𝐸𝛽. Therefore, Γ(𝑀) is a 𝜎(𝐸 ,𝐸 )-closed equicontinuous 󸀠󸀠 󸀠󸀠 󸀠 𝑛 subset of 𝐸 . By Banach-Alaoglu theorem, Γ(𝑀) is 𝜎(𝐸 ,𝐸 )- A polynomial belonging to 𝑃𝑤( 𝐸; 𝐹) is said to be of weak 󸀠󸀠 󸀠 compact. Since 𝑀⊂Γ(𝑀), 𝑀 is also 𝜎(𝐸 ,𝐸 )-compact. type. 󸀠󸀠 (3) It is clear that ⋃𝑀∈B(𝐸) 𝑀⊂𝐸 .Weshallshowthat 󸀠󸀠 󸀠󸀠 󸀠󸀠 Lemma 2. Let 𝐸 and 𝐹 be locally convex spaces and let 𝑢 be 𝐸 ⊂⋃ 𝑀.Let𝑥 be a point of 𝐸 . Then, there exists 𝑛 𝑀∈B(𝐸) an 𝑛-linear mapping belonging to 𝐿𝑤( 𝐸; .Let𝐹) 𝐴1,...,𝐴𝑛 be 𝑉 0 𝐸󸀠 |⟨𝑥, 𝑥󸀠󸀠⟩| ≤ 1 an open neighborhood of in 𝛽 such that absolutely convex bounded subsets of 𝐸.Let𝑎𝑖 be any point of 󸀠 󸀠󸀠 󸀠 for every 𝑥∈𝑉. By the definition of the space 𝐸𝛽 there exists 𝜎(𝐸 ,𝐸 )-closure 𝐴𝑖 of 𝐴𝑖 for each 𝑖 with 2≤𝑖≤𝑛.Wedenote 󸀠󸀠 󸀠󸀠 󸀠 an absolutely convex bounded subset 𝑀∈B(𝐸) such that by N𝑤∗ (𝐸 )(0) the system of all 𝜎(𝐸 ,𝐸 )-neighborhoods of 0 ∘ 󸀠󸀠 󸀠󸀠 𝑀 ⊂𝑉. Thus, by the statement (1) we have in 𝐸 .Then,forany𝛼 ∈ 𝑐𝑠(𝐹) there exists 𝑉∈N𝑤∗ (𝐸 )(0) such that 󸀠󸀠 ∘ ∘∘ 𝑥 ∈𝑉 ⊂𝑀 = 𝑀. (7) 𝛼(𝑢(𝑥1,...,𝑥𝑛)) < 1, (10)

󸀠󸀠 for every (𝑥1,...,𝑥𝑛) ∈ (𝑉∩𝐴1)×((𝑎2 +𝑉)∩𝐴2)×⋅⋅⋅×((𝑎𝑛 + Thus, we have 𝐸 =⋃𝑀∈B(𝐸) 𝑀. 󸀠󸀠 𝑉) ∩ 𝐴𝑛). (4) Since 𝐴 is an equicontinuous subset of 𝐸 , there exists 𝑉 0 𝐸󸀠 𝐴⊂𝑉∘ 𝑉 a neighborhood of in 𝛽 such that .Since is a Proof. Weshallprovethislemmabyinductionon𝑛.Let𝑛= 0 𝐸󸀠 neighborhood of in 𝛽, there exists an absolutely convex 1. Then, the conclusion of this lemma is true since 0∈𝐴1 ∘ 󸀠 󸀠󸀠 󸀠 bounded subset 𝑀 of 𝐸 such that 𝑀 ⊂𝑉.Thus,wehave and 𝜎(𝐸, 𝐸 ) is the induced topology of the topology 𝜎(𝐸 ,𝐸 ) ∘ ∘∘ 𝐴⊂𝑉 ⊂𝑀 .Thiscompletestheproof. onto 𝐸. We suppose that the conclusion of this lemma is true for 𝑛−1 all mappings belonging to 𝐿𝑤( 𝐸; .𝐹) And we assume that AfilterF ={𝐹𝛼} of a locally convex space 𝐸 is called a 𝑛 for 𝑢∈𝐿𝑤( 𝐸; ,𝐹) the conclusion is not true. Then, there Cauchy filter if for every neighborhood 𝑈 of 0 there exists an 󸀠󸀠 exists 𝛼 ∈ 𝑐𝑠(𝐹) such that for every 𝑉∈N𝑤∗ (𝐸 )(0) there 𝐹∈F such that {𝑥−𝑦;𝑥,𝑦∈𝐹}⊂𝑈. A locally convex space is a point: 𝐸 is said to be complete if any Cauchy filter on 𝐸 converges to 𝐸 a point of . There exists the smallest complete locally convex (𝑥1𝑉,𝑥2𝑉,...,𝑥𝑛𝑉)∈(𝑉∩𝐴1)×((𝑎2 +𝑉)∩𝐴2) space 𝐸̃ containing 𝐸 as a subspace. The locally convex space (11) ×⋅⋅⋅×((𝑎 +𝑉)∩𝐴 ), 𝐸̃ is called the completion of 𝐸. 𝑛 𝑛 Abstract and Applied Analysis 3

satisfying for every (𝑧1,𝑧3 ...,𝑧𝑛) ∈ (𝑊(𝑛−1)∩𝐴1)×((𝑎3 +𝑊(𝑛−1))∩ 𝐴3) × ⋅ ⋅ ⋅ × ((𝑎𝑛 +𝑊(𝑛−1))∩𝐴𝑛).Thenwehave 𝛼 (𝑢 (𝑥1𝑉,𝑥2𝑉,...,𝑥𝑛𝑉)) ≥1. (12) 𝛼(𝑢(𝑥1𝑊(𝑛−1),𝑥2𝑊(𝑛−1) −𝑥2𝑊(𝑛−2),...,𝑥𝑛−1𝑊(𝑛−1) 󸀠 Since 𝑢 is 𝜎(𝐸, 𝐸 )-continuous at (0,0,...,0) on each −𝑥 ,𝑥 −𝑥 )) 𝑛 󸀠 𝑛−1𝑊(1) 𝑛𝑊(𝑛−1) 𝑛𝑊 absolutely convex bounded subset of 𝐸 and 𝜎(𝐸, 𝐸 ) is the 󸀠󸀠 󸀠 induced topology of 𝜎(𝐸 ,𝐸 ) onto 𝐸, there exists 𝑊∈ ≥𝛼(𝑢(𝑥1𝑊(𝑛−1),𝑥2𝑊(𝑛−1),...,𝑥𝑛𝑊(𝑛−1))) 󸀠󸀠 N𝑤∗ (𝐸 )(0) such that (21) − ∑ 𝛼(𝑢(𝑥 ,𝑥 ,...,𝑥 )) 1𝑊(𝑛−1) 2𝑊(𝑘2) 𝑛𝑊(𝑘𝑛) 1 (𝑘 )∈𝐾 𝛼 (𝑢 (𝑥 ,𝑥 ,...,𝑥 )) ≤ 𝑖 1 2 𝑛 4 (13) 1 1 ≥1−(2𝑛−1 −1)× ≥ , 2𝑛 2 for every 𝑥𝑖 ∈2(𝑊∩𝐴𝑖) with 1≤𝑖≤𝑛.Wechoosea decreasing sequence 𝑊 ⊃ 𝑊(1) ⊃ 𝑊(2) ⊃ ⋅ ⋅ ⋅ ⊃ 𝑊(𝑛−1) 󸀠󸀠 where 𝐾={(𝑘2,...,𝑘𝑛); 𝑘𝑖 ∈{𝑛−1,𝑛−𝑖},𝑖=2,...,𝑛}\{(𝑛− of elements of N𝑤∗(𝐸 )(0) by the process of (𝑛 − 1) steps as 1,...,𝑛−1)}and 𝑊(0) = 𝑊. follows. If we set At the first step, we consider the (𝑛 − 1)-linear mapping: (𝑦1,𝑦2,...,𝑦𝑛)= (𝑥1𝑊(𝑛−1),𝑥2𝑊(𝑛−1) −𝑥2𝑊(𝑛−2),..., (𝑧1,...,𝑧𝑛−1) 󳨀→ 𝑢 (𝑧1,...,𝑧𝑛−1,𝑥𝑛𝑊) , (14) (22) 𝑥𝑛𝑊(𝑛−1) −𝑥𝑛𝑊), 𝑛−1 belonging to 𝐿𝑤( 𝐸; .𝐹) By the assumption of induction, 󸀠󸀠 we have there exists 𝑊(1) ∈ N𝑤∗ (𝐸 )(0) with 𝑊(1) ⊂ 𝑊 such that 1 𝛼(𝑢(𝑦 ,...,𝑦 )) ≥ , 1 1 𝑛 𝛼(𝑢(𝑧 ,...,𝑧 ,𝑥 )) ≤ 2 1 𝑛−1 𝑛𝑊 𝑛 (15) 2 (23) (𝑦1,𝑦2,...,𝑦𝑛)∈ (𝑊∩𝐴1)×{2(𝑊∩𝐴2)} for every (𝑧1,...,𝑧𝑛−1) ∈ (𝑊(1) ∩ 𝐴1) × ((𝑎2 + 𝑊(1)) ∩ 𝐴2)× ×⋅⋅⋅×{2(𝑊∩𝐴𝑛)} . ⋅ ⋅ ⋅ × ((𝑎𝑛−1 + 𝑊(1)) ∩ 𝐴𝑛−1). (𝑛−1) Atthesecondstep,weconsiderthe -linear mapping This is a contradiction by13 ( ). This completes the proof.

(𝑧1,...,𝑧𝑛−2,𝑧𝑛) 󳨀→ 𝑢 (𝑧1,...,𝑧𝑛−2,𝑥𝑛−1𝑊(1),𝑧𝑛) (16) Lemma 3. Let 𝐸 be a locally convex space, let 𝐹 be a complete 𝑛−1 𝑢 𝑛 belonging to 𝐿𝑤( 𝐸; .𝐹) By the assumption of induction, locally convex space and let be an -linear mapping belonging 󸀠󸀠 𝐿 (𝑛𝐸; 𝐹) 𝐴 ,...,𝐴 there exists 𝑊(2) ∈ N𝑤∗ (𝐸 )(0) with 𝑊(2) ⊂ 𝑊(1) such to 𝑤 .Let 1 𝑛 be absolutely convex bounded 𝐸 𝜎(𝐸󸀠󸀠,𝐸󸀠) that subsets of . Then there exists a -continuous mapping 𝑢̃𝐴 ×⋅⋅⋅×𝐴 from 𝐴1 ×⋅⋅⋅×𝐴𝑛 into 𝐹 such that 𝑢̃𝐴 ×⋅⋅⋅×𝐴 =𝑢on 1 𝐴 1 ×⋅⋅⋅×𝐴𝑛 1 𝑛 𝛼(𝑢(𝑧 ,...,𝑧 ,𝑥 ,𝑧 )) ≤ 1 𝑛. 1 𝑛−2 𝑛−1𝑊(1) 𝑛 2𝑛 (17) 󸀠󸀠 󸀠 Proof. Let 𝑎𝑖 be any point of the 𝜎(𝐸 ,𝐸 )-closure 𝐴𝑖 of 𝐴𝑖 for for every point (𝑧1,...,𝑧𝑛−2,𝑧𝑛) of each 𝑖 with 1≤𝑖≤𝑛. At first, we shall show that a filter of 𝐹 F (𝑎 ,...,𝑎 )= {𝑢((𝑎 +𝑉)∩𝐴 ,...,(𝑎 +𝑉)∩𝐴 ); (𝑊 (2) ∩𝐴1)×((𝑎2 +𝑊(2))∩𝐴2) 1 𝑛 1 1 𝑛 𝑛 󸀠󸀠 ×⋅⋅⋅×((𝑎𝑛−2 +𝑊(2))∩𝐴𝑛−2) × ((𝑎𝑛 +𝑊(2))∩𝐴𝑛). 𝑉∈N𝑤∗ (𝐸 ) (0)} (18) (24) Repeating this process, at the (𝑛 − 1)th step, we consider (𝑛 − 1) is a Cauchy filter. Let 𝛼 be an arbitrary continuous seminorm the -linear mapping: 󸀠󸀠 of 𝐹.ByLemma 2 there exists 𝑉∈N𝑤∗ (𝐸 )(0) such that (𝑧 ,𝑧 ,...,𝑧 )󳨀→𝑢(𝑧,𝑥 ,𝑧 ,...,𝑧 ), 1 3 𝑛 1 2𝑊(𝑛−2) 3 𝑛 (19) 1 𝛼 (𝑢 (𝑥1,𝑥2,...,𝑥𝑛)) < , (25) 𝑛−1 𝑛 belonging to 𝐿𝑤( 𝐸; .𝐹) By the assumption of induction, 󸀠󸀠 there exists 𝑊(𝑛−1) ∈ N𝑤∗ (𝐸 )(0) with 𝑊(𝑛−1) ⊂ 𝑊(𝑛−2) for every point (𝑥1,𝑥2,...,𝑥𝑛) of such that 𝑛 𝑛 1 ⋃ (∏ ((𝛾 𝑎 +2𝑉)∩2𝐴 )) , 𝛼(𝑢(𝑧 ,𝑥 ,𝑧 ,...,𝑧 )) ≤ 𝑖𝑗 𝑗 𝑗 (26) 1 2𝑊(𝑛−2) 3 𝑛 2𝑛 (20) 𝑖=1 𝑗=1 4 Abstract and Applied Analysis

𝑛 󸀠󸀠 󸀠󸀠 󸀠 where 𝛾𝑖𝑗 =1(𝑖=𝑗)̸ and 𝛾𝑖𝑗 = 0 (𝑖 = 𝑗).Then,wehave (3) There exists 𝑝∈𝑃̃ 𝑎( 𝐸 ;𝐹)such that 𝑝̃ is 𝜎(𝐸 ,𝐸 )- 󸀠󸀠 continuous on each equicontinuous subset of 𝐸 and 𝛼(𝑢(𝑥1,𝑥2,...,𝑥𝑛)−𝑢(𝑦1,𝑦2,...,𝑦𝑛)) 𝑝|𝐸=𝑝̃ . 𝑝 ≤ 𝛼 (𝑢1 (𝑥 −𝑦1,𝑥2,...,𝑥𝑛)) (4) is weakly uniformly continuous on every bounded subset of 𝐸. +𝛼(𝑢(𝑦1,𝑥2 −𝑦2,𝑥3,...,𝑥𝑛)) Proof. We shall show that (1) implies (3). There exists a (27) 𝑛 +𝛼(𝑢(𝑦1,𝑦2,𝑥3 −𝑦3,𝑥4,...,𝑥𝑛)) symmetric 𝑛-linear mapping 𝑢 from 𝐸 into 𝐹 such that 𝑝(𝑥) = 𝑢(𝑥,...,𝑥) for every 𝑥∈𝐸.Bythepolarization +⋅⋅⋅+𝛼(𝑢(𝑦,𝑦 ,...,𝑦 ,𝑥 −𝑦 )) 𝑛 1 2 𝑛−1 𝑛 𝑛 formula, we have 𝑢∈𝐿𝑤( 𝐸, .By𝐹) Lemma 4 there exists 𝑢∈𝐿̃ (𝑛𝐸󸀠󸀠;𝐹) 𝑢|𝐸̃ 𝑛 =𝑢 𝑢̃ 𝜎(𝐸󸀠󸀠,𝐸󸀠) 1 𝑎 with such that is - ≤𝑛× =1, continuous on 𝐴1 ×⋅⋅⋅×𝐴𝑛 for all absolutely convex bounded 𝑛 𝑛 󸀠󸀠 subsets 𝐴1,...,𝐴𝑛 of 𝐸.Wedefine𝑝∈𝑃̃ 𝑎( 𝐸 ;𝐹)by 𝑝(𝑥)̃ = 𝑛 𝑢(𝑥,...,𝑥)̃ 𝑥∈𝐸󸀠󸀠 𝑝̃ for every (𝑥1,...,𝑥𝑛), (𝑦1,...,𝑦𝑛)∈∏𝑖=1((𝑎𝑖 +𝑉)∩𝐴𝑖).Thus, for every .ByLemma 1, satisfies all (3) the filter F(𝑎1,...,𝑎𝑛) is a Cauchy filter. Since 𝐹 is complete required conditions of the statement . ∘∘ and Hausdorff, there exists uniquely the limit point of the (3) implies (2) since 𝑀 is equicontinuous for every filter F(𝑎1,...,𝑎𝑛) for every (𝑎1,...,𝑎𝑛)∈𝐴1 ×⋅⋅⋅×𝐴𝑛. absolutely convex bounded subset 𝑀 of 𝐸. 𝑢̃ (𝑎 ,...,𝑎 ) We shall show that (2) implies (4).Let𝐴 be a bounded We denote by 𝐴1×⋅⋅⋅×𝐴𝑛 1 𝑛 the limit point of the filter subset of 𝐸. We denote by 𝑀 theabsolutelyconvexhull F(𝑎1,...,𝑎𝑛) for every (𝑎1,...,𝑎𝑛)∈𝐴1 × ⋅⋅⋅ × 𝐴𝑛.Then 󸀠󸀠 󸀠 of 𝐴 in 𝐸.Then,𝑀 is a bounded subset of 𝐸,and𝐴⊂ 𝑢̃𝐴 ×⋅⋅⋅×𝐴 defines 𝜎(𝐸 ,𝐸 )-continuous mapping from 𝐴1 × 󸀠󸀠 󸀠 1 𝑛 𝑀.Bystatement(2), there exists a 𝜎(𝐸 ,𝐸 )-continuous ⋅⋅⋅×𝐴𝑛 into 𝐹 with ̃ ̃ mapping 𝑝𝑀 from 𝑀 into 𝐹 such that 𝑝𝑀 =𝑝on 𝑀. 󸀠󸀠 󸀠 Since by Lemma 1 𝑀 is 𝜎(𝐸 ,𝐸 )-compact, 𝑝̃ is uniformly 𝑢̃𝐴 ×⋅⋅⋅×𝐴 =𝑢 on 𝐴1 ×⋅⋅⋅×𝐴𝑛. (28) 𝑀 1 𝑛 󸀠󸀠 󸀠 ̃ 𝜎(𝐸 ,𝐸 )-continuous on 𝑀.Since𝑝𝑀 =𝑝on 𝐴, 𝑝 is weakly This completes the proof. uniformly continuous on 𝐴. This implies (4).Itisclearthat (4) implies (1).Thiscompletestheproof. 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 𝑛 𝑛 󸀠󸀠 𝑛 3. The Tensor Product Representation of to 𝐿𝑤( 𝐸; .𝐹) Then, there exists 𝑢∈𝐿̃ 𝑎( 𝐸 ;𝐹)with 𝑢|𝐸̃ =𝑢 󸀠󸀠 󸀠 such that 𝑢̃ is 𝜎(𝐸 ,𝐸 )-continuous on 𝐴1 × ⋅⋅⋅ × 𝐴𝑛 for all Polynomials in Locally Convex Spaces absolutely convex bounded subsets 𝐴1,...,𝐴𝑛 of 𝐸. 𝑛 For any 𝑢∈𝐿𝑎( 𝐸; ,weset𝐹) Proof. By Lemma 3 for all absolutely convex bounded subsets 1 𝐴 ,...,𝐴 𝐸 𝜎(𝐸󸀠󸀠,𝐸󸀠) 𝑠 (𝑢) (𝑥 ,...,𝑥 )= ∑ 𝑢(𝑥 ,...,𝑥 ) 1 𝑛 of , there exists a -continuous mapping 1 𝑛 𝑛! 𝜎(1) 𝜎(𝑛) (30) 𝑢̃ 𝐴 ×⋅⋅⋅×𝐴 𝐹 𝑢̃ =𝑢 𝐴 ×⋅⋅⋅× 𝜎∈𝑆𝑛 𝐴1×⋅⋅⋅×𝐴𝑛 from 1 𝑛 to with 𝐴1×⋅⋅⋅×𝐴𝑛 on 1 𝐴 𝐴 ,...,𝐴 ,𝐵 ,...,𝐵 𝑛.If 1 𝑛 1 𝑛 are absolutely convex bounded 𝑛 for every (𝑥1,...,𝑥𝑛)∈𝐸,where𝑆𝑛 is the permutation group subsets of 𝐸 with 𝐴𝑖 ∩ 𝐵𝑖 =0̸ for 1≤𝑖≤𝑛,wehave of degree 𝑛.Then,𝑠(𝑢) is a symmetric 𝑛-linear mapping from 𝑛 𝑢̃ = 𝑢̃ 𝐸 to 𝐹 satisfying 𝐴1×⋅⋅⋅×𝐴𝑛 𝐵1×⋅⋅⋅×𝐵𝑛 (29) 𝑢 (𝑥,...,𝑥) =𝑠(𝑢)(𝑥,...,𝑥) on (𝐴1 ×⋅⋅⋅×𝐴𝑛)∩(𝐵1 ×⋅⋅⋅×𝐵𝑛).Thus,byLemma 1,we (31) 󸀠󸀠𝑛 can define an n-linear mapping 𝑢̃ from 𝐸 into 𝐹 by setting 𝑠 𝑛 for every 𝑥∈𝐸. We denote 𝐿𝑎( 𝐸; 𝐹) by the space of all 𝑢=̃ 𝑢̃𝐴 ×⋅⋅⋅×𝐴 on 𝐴1 × ⋅⋅⋅ × 𝐴𝑛 for all absolutely convex 𝑛 1 𝑛 symmetric 𝑛-linear mappings from 𝐸 to 𝐹.LetΔ 𝑛 be the bounded subsets 𝐴1,...,𝐴𝑛 of 𝐸.Then,themapping𝑢̃ 𝑛 mapping from 𝐸 into 𝐸 defined by satisfies all required conditions of this lemma. This completes the proof. Δ 𝑛 (𝑥) = (𝑥,...,𝑥) for every 𝑥∈𝐸. (32)

The following theorem is proved by Aron et al. [4], 𝑛 For any 𝑢∈𝐿𝑎( 𝐸; ,𝐹) we define an 𝑛-homogeneous Gonzalez´ and Gutierrez´ [5], Honda et al. [6]. ∗ polynomial Δ 𝑛(𝑢) by Theorem 5. Let 𝐸 be a locally convex space and let 𝐹 be a 𝑛 ∗ complete locally convex space. Let 𝑝∈𝑃(E;𝐹).Then,the Δ 𝑛 (𝑢) =𝑢∘Δ𝑛. (33) following statements are equivalent. 𝑛 The mapping (1) 𝑝∈𝑃𝑤( 𝐸; .𝐹) ∗ 𝑠 𝑛 𝑛 (2) For each absolutely convex bounded subset 𝑀 of 𝐸, 𝑝 Δ 𝑛 :𝐿𝑎 ( 𝐸; 𝐹)𝑎 󳨀→𝑃 ( 𝐸; 𝐹) (34) 󸀠󸀠 󸀠 ∘∘ can be extended 𝜎(𝐸 ,𝐸 )-continuously to 𝑀 ,where ∘∘ 󸀠󸀠 󸀠 𝑀 is the bipolar set of 𝑀 for the dual pair ⟨𝐸 ,𝐸 ⟩. is surjective. Abstract and Applied Analysis 5

𝑛 Theorem 6 (polarization formula). Let 𝑝∈𝑃𝑎( 𝐸; 𝐹) and let For any 𝑥1 ⊗⋅⋅⋅⊗𝑥𝑛 ∈ ⨂𝑛𝐸,weset 𝑠 𝑛 ∗ 𝑢∈𝐿𝑎( 𝐸; .If𝐹) Δ 𝑛(𝑢) =,then 𝑝 1 𝑠(𝑥1 ⊗⋅⋅⋅⊗𝑥𝑛)= ∑ 𝑥𝜎(1) ⊗⋅⋅⋅⊗𝑥𝜎(𝑛). 𝑛 𝑛! (42) 1 𝜎∈𝑆𝑛 𝑢 (𝑥 ,...,𝑥 ) = ∑ 𝜖 ⋅⋅⋅𝜖 𝑝 (∑𝜖 𝑥 ) . 1 𝑛 2𝑛𝑛! 1 𝑛 𝑖 𝑖 (35) ⨂ 𝐸 ⨂ 𝐸 𝜖𝑖=±1 𝑖=1 We denote by 𝑛,𝑠 the subspace of 𝑛 generated by 𝑠(𝑥1 ⊗⋅⋅⋅⊗𝑥𝑛), 𝑥𝑖 ∈𝐸.Thespace⨂𝑛,𝑠𝐸 is called the 𝑛-fold ∗ 𝐸 ⨂ 𝐸 By the above polarization formula, the mapping Δ 𝑛 : symmetric tensor product space of . Elements of 𝑛,𝑠 are 𝑠 𝑛 𝑛 𝑛 𝐿𝑎( 𝐸; 𝐹)𝑎 →𝑃 ( 𝐸; 𝐹) is a linear isomorphism. called -symmetric tensors. Clearly, every tensor of the form 𝑥⊗⋅⋅⋅⊗𝑥 We denote by ⨂𝑛𝐸 the 𝑛-fold tensor product space of 𝐸. is a symmetric tensor. Moreover, each element 𝑛 𝜃 ⨂ 𝐸 Let 𝑖𝑛 be the linear mapping from 𝐸 into ⨂𝑛𝐸 defined by in 𝑛,𝑠 can be expressed as a finite sum (not necessarily unique) of the form 𝑖𝑛 (𝑥1,...,𝑥𝑛)=𝑥1 ⊗⋅⋅⋅⊗𝑥𝑛 (36) ∑𝑥 ⊗⋅⋅⋅⊗𝑥. 𝑖 𝑖 (43) 𝑛 𝑛 𝑖 for every (𝑥1,...,𝑥𝑛)∈𝐸.Forany𝑢∈𝐿𝑎( 𝐸; ,𝐹) there exists 𝑖∗(𝑢) ∈ 𝐿 (⨂ 𝐸; 𝐹) 𝑛 ∗ aunique 𝑛 𝑎 𝑛 such that the diagram For any 𝑝∈𝑃𝑎( 𝐸; ,𝐹) there exists a unique 𝑗𝑛 (𝑝) ∈ 𝐿𝑎(⨂𝑛,𝑠𝐸; 𝐹) such that the diagram u p En F E F

∗ (37) in i (u) (44) n ∗ 𝛿n jn (p) ⨂ E n ⨂ E n,s commutes. The mapping commutes. The mapping

𝑖∗ :𝐿 (𝑛𝐸; 𝐹) 󳨀→𝐿 (⨂𝐸; 𝐹) ∗ 𝑛 𝑛 𝑎 𝑎 (38) 𝑗𝑛 :𝑃𝑎 ( 𝐸; 𝐹) 󳨀→𝐿𝑎 (⨂𝐸; 𝐹) (45) 𝑛 𝑛,𝑠 𝑛 𝑛 𝑠 𝑛 is a linear isomorphism. Each element of ⨂𝑛𝐸 has a repre- is a linear isomorphism. Let 𝑃( 𝐸; ,𝐹) 𝐿( 𝐸; ,and𝐹) 𝐿 ( 𝐸; 𝐹) sentation of the form be, respectively, the spaces of continuous 𝑛-homogeneous 𝐸 𝐹 𝑛 ℓ polynomials from into and the continuous symmetric - linear mappings from 𝐸 into 𝐹. The restrictions ∑𝑥𝑖,1 ⊗𝑥𝑖,2 ⊗⋅⋅⋅⊗𝑥𝑖,𝑛. (39) ∗ 𝑛 𝑛 𝑖=1 Δ 𝑛 :𝐿(𝐸; 𝐹) 󳨀→ 𝑃( 𝐸; 𝐹) , ∗ 𝑠 𝑛 𝑛 However, this representation will never be unique. We Δ 𝑛 :𝐿 ( 𝐸; 𝐹) 󳨀→ 𝑃( 𝐸; 𝐹) , (46) denote by 𝛿𝑛 the mapping of 𝐸 into ⨂𝑛𝐸 defined by 𝑠:𝐿(𝑛𝐸; 𝐹)𝑠 󳨀→𝐿 (𝑛𝐸; 𝐹) 𝛿𝑛 (𝑥) =𝑥⊗⋅⋅⋅⊗𝑥 (40) are well-defined. For each 𝛼 ∈ 𝑐𝑠(𝐸) and 𝜃=∑𝑖 𝑥𝑖 ⊗⋅⋅⋅⊗𝑥𝑖 ∈ for every 𝑥∈𝐸. ⨂𝑛,𝑠𝐸,weset

Proposition 7. Amapping𝑝:𝐸is →𝐹 an 𝑛-homogeneous 𝑛 𝑛 𝜋𝛼,𝑛 (𝜃) = inf {∑𝛼(𝑥𝑖) |𝜃=∑𝑥𝑖 ⊗⋅⋅⋅⊗𝑥𝑖}. (47) polynomial if and only if there exists 𝑇∈𝐿𝑎( 𝐸; 𝐹) such that 𝑖 𝑖 the diagram 𝜋𝛼,𝑛 is a seminorm on ⨂𝑛,𝑠𝐸. We define the 𝜋-topology Δ or the projective topology on ⨂ 𝐸 as the locally convex E n En 𝑛,𝑠 topology generated by {𝜋𝛼,𝑛}𝛼∈𝑐𝑠(𝐸).Wedenoteby⨂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 p [2]). 𝛿n u (41) Proposition 8. Let 𝐸 be a locally convex space, then we have

𝐿(𝑛𝐸; 𝐹)⨂ ≅𝐿( 𝐸; 𝐹) , 𝑛.𝜋 ⨂ E F (48) n T 𝐿𝑠 (𝑛𝐸; 𝐹) ≅𝐿(⨂𝐸; 𝐹) ≅𝑃(𝑛𝐸; 𝐹) . commutes. 𝑛.𝑠,𝜋 6 Abstract and Applied Analysis

󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠 Let E(𝐸 ) be the family of all equicontinuous subsets of An 𝑛-homogeneous polynomial 𝑝 on 𝐸 is 𝜎(𝐸 ,𝐸 )- 󸀠󸀠 󸀠󸀠 󸀠 󸀠󸀠 𝐸 with respect to the dual pair ⟨𝐸 ,𝐸 ⟩. We denote by O𝑤∗,𝜖 continuous on every equicontinuous subsets of 𝐸 if and only 󸀠󸀠 󸀠󸀠 the family of subsets of 𝐸 defined by if 𝑝 is 𝜏𝑤∗,𝜖-continuous on 𝐸 .Thus,byPropositions9 and 11, we have the following theorem. 󸀠󸀠 O𝑤∗,𝜖 = {𝑉|𝑉⊂𝐸 ,𝑉∩𝑀are Theorem 13. If an 𝑛-homogeneous polynomial 𝑝 on 𝐸 is of 󸀠󸀠 󸀠 󸀠󸀠 𝑝 𝜏 𝐸 𝜎(𝐸 ,𝐸 ) -open in 𝑀 for all 𝑀∈E (𝐸 )}. weaktypeifandonlyif is 𝑜-continuous in and continuous on 𝐸 with the initial topology of 𝐸.

(49) Then, we can obtain the following topological tensor 󸀠󸀠 productrepresentationofpolynomialsofweaktype. We denote by 𝜏𝑤∗,𝜖 the topology on 𝐸 such that the family of all 𝜏𝑤∗,𝜖-open sets coincides with O𝑤∗,𝜖.ByTheorem 5,the Theorem 14. Let 𝐸 be a DF space, then we have following is valid. 𝐿(⨂𝐸 ;𝐹)∩𝐿(⨂𝐸; )≅𝑃 (𝑛𝐸; 𝐹) Proposition 9. 𝑛 𝑝 𝐸 𝜏0 F 𝑤 (53) A -homogeneous polynomial on is of 𝑛,𝑠𝜋 𝑛,𝑠𝜋 weak type if and only if there exists a 𝜏𝑤∗,𝜖-continuous 𝑛- ̃ 󸀠󸀠 ̃ homogeneous polynomial 𝑝 on 𝐸 such that 𝑝=𝑝on 𝐸. for every complete locally convex space 𝐹.

𝜏 ∗ However, in general, the topology 𝑤 ,𝛽 is not a locally A topological space 𝑋 is called a 𝑘-space if its topology convex topology (cf. Komura¯ [7]). is localized on its compact set;, that is, 𝑈⊂𝑋is open if and only if 𝑈∩𝐾is open in 𝐾,withtheinducedtopology,for 𝐸 Definition 10. A locally convex space is called a DF-space if each compact subset 𝐾 of 𝑋.Amappingfroma𝑘-space into it contains a countable fundamental system of bounded sets a topological space is continuous if and only if its restriction and if the intersection of any sequence of absolutely convex to each compact set is continuous. neighborhoods of 0 which absorbs all bounded sets is itself a Let 𝑢 beamappingfromalocallyconvexspace𝐸 into neighborhood of 0. a locally convex space 𝐹.If𝐸 is a 𝑘-space and if 𝑢 is 󸀠 𝜎(𝐸, 𝐸 )-continuousoneveryboundedsubsetof𝐸,then𝑢 Grothendieck [8, 9] proved that the strong dual space of a is continuous on 𝐸 with the initial topology of 𝐸.Thus,we metrizable locally convex space is a DF-space and the strong obtain the following theorem. dual space of a DF-space is a Frechet´ space. All Banach spaces are DF-spaces. The following result is Theorem 15. If 𝐸 is a DF space and a 𝑘-space, then we have known. Proposition 11 𝐸 𝐿(⨂𝐸 ;𝐹)≅𝑃 (𝑛𝐸; 𝐹) (Banach-Dieudonnetheorem).´ Let be a 𝜏0 𝑤 (54) 󸀠 𝑛,𝑠𝜋 metrizable locally convex space. Then, the topology 𝜏𝑤∗,𝜖 on 𝐸 is the topology of uniform convergence on all compact sets in 𝐸. for every complete locally convex space 𝐹. Proof. The proof is on Kothe¨ [10,§21–10]. If 𝐸 is a Banach space, then 𝐸 is a DF-space and a 𝑘-space. 󸀠 Thus, we have the following corollary. If 𝐸 is a DF space, then 𝐸𝛽 is a Frechet´ space. Therefore, by Proposition 11 the following is valid. Corollary 16. If 𝐸 is a Banach space, then we have 󸀠󸀠 Proposition 12. The topology 𝜏𝑤∗,𝜖 on 𝐸 is the topology of 𝐸󸀠 𝐿(⨂𝐸 ;𝐹)≅𝑃 (𝑛𝐸; 𝐹) uniform convergence on all compact sets in . 𝜏0 𝑤 (55) 𝑛,𝑠𝜋 󸀠󸀠 We denote by 𝛼𝐾 the seminorm on 𝐸 defined by for every complete locally convex space 𝐹. 󵄨 󵄨 󸀠󸀠 󵄨 󸀠󸀠 󸀠 󵄨 󸀠 𝛼𝐾 (𝑥 )=sup {󵄨⟨𝑥 ,𝑥 ⟩󵄨 |𝑥 ∈𝐾} (50) Conflict of Interests 󸀠 for every compact subsets 𝐾 of 𝐸 . We denote by 𝜏0 the locally 𝛽 The authors declare that there is no conflict of interests 𝐸󸀠󸀠 convex topology on defined by the set of seminorms: regarding the publication of this paper. 󸀠 {𝛼𝐾 |𝐾is every compact set of 𝐸 }. (51) Acknowledgments

We denote by 𝐸𝜏 the locally convex space 𝐸 defined the set 0 KwangHoShonwassupportedbyBasicScienceResearch of seminorms: Program through the National Research Foundation of Korea {𝛼 | |𝐾 𝐸󸀠}. (NRF) funded by the Ministry of Science and ICT and Future 𝐾 𝐸 is every compact set of (52) Planning (2013R1A1A2008978). Abstract and Applied Analysis 7

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