aa r X i v : . [ m a t h . F A ] O c t The Goldstine-Weston theorem in random normedmodules
Guang Shi LMIB and School of Mathematics and Systems Science, Beihang University, Beijing100191, P.R. China
Abstract
This article generalize the classical Goldstine-Weston theorem on normedspaces to one on random normed modules: the image of a random normedmodule ( E, k·k ) under the random natural embedding J is dense in its doublerandom conjugate space E ∗∗ with respect to the ( ε, λ ) weak star topology;and J ( E ) is also dense in E ∗∗ with respect to the locally L -convex weak startopology if E has the countable concatenation property. Keywords:
Random normed modules, Hyperplane separation theorem, Randomconjugate spaces, ( ε, λ ) weak star topology, Locally L -convex weak startopology
1. Introduction
Random normed modules (briefly, RN modules) is proved to be a properand powerful tool in the study of conditional risk measures[12, 13]. This suc-cessful application partly attributes to the systematic and deep developmentof the theory of RN modules and random conjugate spaces[2, 3, 5, 6, 7, 8,9, 10]. As pointed out in [5], the structure of an RN module is an organiccombination of the structure of a normed space with the structure of a mea-surable space. As a result, some classical theorem on normed spaces may stillhold by the RN modules, like James theory[10]; and others may not holduniversally for random normed modules unless they possess extremely simplestratification structure, like the Banach-Alaoglu theorem[5]. Exploring how Supported by NNSF No. 10871016
Preprint submitted to Elsevier November 16, 2018 he classical properties of normed spaces are preserved in RN modules isimportant for further study and application of RN modules.The purpose of this paper is to establish similar results in RN modulescorresponding to the classical Goldstine-Weston theorem in normed spaces.Our results shows that the random weak star topologies on RN modulesprocess many similar properties to the classical weak star topology. Besides,one could also see the Goldstine-Weston theorem in RN modules is exactlythe same as the classical one under the ( ε, λ ) topology; and slightly differentfrom the classical one under the locally L -convex topology. This is due tothat the ( ε, λ ) topology is very natural and the locally L -convex topology isvery strong.The key step in our approach is that we generalize the classical Hellytheorem characterizing the existence of solution for linear equations on anormed space to one characterizing the existence of solution for random linearequations on a random normed module. On one hand, we make full use of therecently developed theory of RN modules in the prove of the Helly theoremin RN modules; and on the other hand, the establishment of our final resultdepends much more on the Helly theorem in RN modules than that in thenormed spaces since the Banach-Alaoglu theorem is not universally availablein RN modules as mentioned above.The paper proceeds as follows: section 2 gives some necessary notions andpreliminaries. In section 3, we proved the helly theorem in random normedmodules. And in section 4, we state and prove our main results.
2. Preliminaries
Throughout this paper, (Ω , F , P ) denotes a probability space, N denotesthe set of positive integers, and K the scalar field R of real numbers or C of complex numbers. Let L ( F , R ) be the set of equivalence classes ofreal-valued F –measurable random variables on Ω. Define the ordering on L ( F , ¯ R ) by ξ η iff ξ ( ω ) η ( ω ) for P –almost all ω in Ω (briefly, a.s.),where ξ and η are arbitrarily chosen representatives of ξ and η , respectively.It is well known from [11] that L ( F , R ) is a complete lattice in the sensethat every subset with an upper bound has a supremum. If A is a subsetwith an upper bound of L ( F , R ), we denote the supremum of A by ∨ A .Besides, for any A ∈ F and ξ, η ∈ L ( F , ¯ R ), “ ξ > η on A ” means ξ ( ω ) > η ( ω ) a.s. on A for any chosen representative ξ and η of ξ and η ,2espectively. As usual, ξ > η means ξ > η and ξ = η ; and [ ξ > η ] denotes theequivalence class of the F -measurable set { ω ∈ Ω | ξ ( ω ) > η ( ω ) } . More-over, L = { ξ ∈ L ( F , R ) | ξ > } and L = { ξ ∈ L ( F , R ) | ξ > } .And in the sequel of this paper we make the following convention: if I A denotes the characteristic function of an F –measurable set A , then we use˜ I A for its equivalence class in L ( F , K ). Besides, for any ξ ∈ L ( F , K ), | ξ | and ξ − respectively stand for the equivalence classes determined by the F -measurable function | ξ | : Ω → R defined by | ξ | ( ω ) = | ξ ( ω ) | , ∀ ω ∈ Ω and( ξ ) − defined by ( ξ ) − ( ω ) = (cid:26) ( ξ ( ω )) − , ξ ( ω ) = 0;0 , otherwise , where ξ is an arbitrarily chosen representative of ξ . It is clear that | ξ | ∈ L and ξ · ξ − = ˜ I { ω ∈ Ω | ξ ( ω ) =0 } . Definition 2.1 [1].
Let E be a left module over the algebra L ( F , K ).A countable concatenation of some sequence { x n | n ∈ N } in E with re-spect to some countable partition { A n | n ∈ N } of Ω is a formal sumΣ n ∈ N ˜ I A n x n . Moreover, a countable concatenation Σ n ∈ N ˜ I A n x n is well definedor Σ n ∈ N ˜ I A n x n ∈ E if there is x ∈ E such that ˜ I A n x = ˜ I A n x n , ∀ n ∈ N . Asubset A of E is called having the countable concatenation property if everycountable concatenation Σ n ∈ N ˜ I A n x n with x n ∈ A for each n ∈ N still belongsto A , namely Σ n ∈ N ˜ I A n x n is well defined and there exists x ∈ A such that x = Σ n ∈ N ˜ I A n x n . And for a subset M of E , H cc ( M ) denotes the countablecatenation hull of M .Suppose M and G are two nonempty subsets of an L ( F , K )-module E .If M and G have the countable concatenation property and M ∩ G = ∅ ,then H ( M, G ) denotes the hereditarily disjoint stratification of M and G [1,Definition 3.14]. Definition 2.2 [1, 4].
An ordered pair ( E, k · k ) is called a random normedmodule over K with base (Ω , F , P ) if E is a left module over the algebra L ( F , K ) and k · k is a mapping from E to L such that the following threeaxioms are satisfied:(1) k x k = 0 iff x = θ (the null element of E );32) k ξx k = | ξ |k x k , ∀ ξ ∈ L ( F , K ) and ∀ x ∈ E ;(3) k x + y k k x k + k y k , ∀ x, y ∈ E . Definition 2.3 [1, 4].
An ordered pair ( E, h· , ·i ) is called a random innerproduct module (briefly, an RIP module) over K with base (Ω , F , P ) if E is a left module over the algebra L ( F , K ) and h· , ·i : E × E → L ( F , K )satisfies the following statements:(1) h x, x i ∈ L and h x, x i = 0 iff x = θ ;(2) h x, y i = h y, x i , ∀ x, y ∈ E where h y, x i denotes the complex conjugateof h y, x i ;(3) h ξx, y i = ξ h x, y i , ∀ ξ ∈ L ( F , K ) and ∀ x, y ∈ E ;(4) h x + y, z i = h x, z i + h y, z i , ∀ x, y, z ∈ E .where h x, y i is called the random inner product between x and y .An RIP module ( E, h· , ·i ) is also an RN module when k · k : E → L isdefined by k x k = p h x, x i , ∀ x ∈ E . Example 2.4.
Denote by L ( F , K n ) the linear space of equivalence classesof K n –valued F –measurable functions on Ω, where n is a positive integer. De-fine · : L ( F , K ) × L ( F , K n ) → L ( F , K n ) by λ · x = ( λξ , λξ , · · · , λξ n ) and h· , ·i : L ( F , K n ) × L ( F , K n ) → L ( F , K ) by h x, y i = Σ ni =1 ξ i ¯ η i ,for any λ ∈ L ( F , K ) and x = ( ξ , ξ , · · · , ξ n ), y = ( η , η , · · · , η n ) ∈ L ( F , K n ). It is easy to check that ( L ( F , K n ) , h· , ·i ) is an RIP moduleover K with base (Ω , F , P ), and also an RN module. Specially, L ( F , K ) isan RN module and k λ k = | λ | for any λ ∈ L ( F , K ). Definition 2.5 [4].
Let ( E, k · k ) be an RN module over K with base(Ω , F , P ). Then a linear operator f from E to L ( F , K ) is called a P -a.e.bounded random linear functional on E if there exists some ξ in L suchthat | f ( x ) | ξ · k x k , ∀ x ∈ S .Denote by E ∗ the linear space of all P -a.e. bounded random linear func-tionals on an RN module ( E, k · k ) over K with base (Ω , F , P ). Define k · k ∗ : E ∗ → L by k f k ∗ = ∨{| f ( y ) | | y ∈ E and k y k , } and · : L ( F , K ) × E ∗ → E ∗ by ( ξ · f )( x ) = ξ · ( f ( x )), ∀ ξ ∈ L ( F , K ), ∀ f ∈ E ∗ ∀ x ∈ E . Then it is easy to check that ( E ∗ , k · k ∗ ) is an RN module over K with base (Ω , F , P ). Such ( E ∗ , k · k ∗ ) is called the random conjugate spaceof ( E, k · k )[4].Finally, this section is ended by introducing some useful topologies definedon RN modules and their random conjugate spaces. A natural topology foran RN modules ( E, k · k ) over K with base (Ω , F , P ) is the ( ε, λ )-topology[4],which is denoted by T ε,λ in this paper. A subset A of E is T ε,λ open iff foreach x ∈ E , there exist two positive real numbers ε, λ such that λ < N x ( ε, λ ) = { y ∈ E | P ([ k x − y k < ε ]) > − λ } is included in A . And anotherstronger topology on E is the locally L -convex topology[12], which is denotedby T c in this paper. A subset B is T c open iff for each x ∈ B there exists ǫ ∈ L such that N x ( ε ) = { y ∈ E | k y − x k < ǫ } is included in B . Furthermore,similarly to the classical conjugate spaces, there are another two topologiesfor the random conjugate spaces ( E ∗ , k · k ∗ ) of E besides the above twotopologies, namely the ( ε, λ ) weak star topology(denoted by σ ( ε,λ ) ( E ∗ , E ))and the locally L -convex weak star topology(denoted by σ c ( E ∗ , E )). If θ ∗ is the null element of E ∗ , then typical neighborhood systems of the null el-ement θ ∗ of E ∗ in σ ( ε,λ ) ( E ∗ , E ) and σ c ( E ∗ , E ) are the collection of the set N θ ∗ ( x , x , · · · , x n , ε, λ ) = { g ∈ E ∗ | P ([ | g ( x i ) | < ε ]) > − λ, i n } andthe collection of N θ ∗ ( x , x , · · · , x n , ǫ ) = { g ∈ E ∗ | | g ( x i ) | < ǫ, i n } forall n ∈ N , x , x , · · · , x n ∈ E , ε >
0, 0 < λ < ǫ ∈ L . For details ofthese two topologies, and also terminologies such as L -convex, L -absorbentand L -balanced, we refer the readers to [1, 12].
3. The Helly Theorem in random normed modulesLemma 3.1.
Let ( E, k · k ) be an RN module over R with base (Ω , F , P ) .Suppose G and M are two nonempty L -convex subsets of E with the count-able concatenation property and the T c interior G ◦ of G is not empty. If G ∩ M = ∅ , then there exists f ∈ E ∗ such that f ( x ) f ( y ) on H ( G, M ) for all x ∈ G and y ∈ M and f ( x ) < f ( y ) on H ( G, M ) for all x ∈ G ◦ and y ∈ M .If R is replaced by C , then the above statements still hold in the followingway: ( Ref )( x ) ( Ref )( y ) on H ( G, M ) for all x ∈ G and y ∈ M nd ( Ref )( x ) < ( Ref )( y ) on H ( G, M ) for all x ∈ G ◦ and y ∈ M .Here H ( G, M ) denotes the hereditarily disjoint stratification of H and M ,and ( Ref )( x ) = Re ( f ( x )) , ∀ x ∈ E . Proof.
By [1, Theorem 3.13] it follows that P ( H ( G, M )) >
0. First supposethat H ( G, M ) = Ω.Let A = M − G = { y − x | x ∈ G and y ∈ M } . Clearly the T c interior A ◦ of A is nonempty. Define B = { z − x | x ∈ A } for some fixed z ∈ A ◦ .It follows that B is an L -convex subset of E ; and B is also L -absorbentsince B contains a T c -neighborhood of θ , where θ is the null element of E .Moreover, it is easy to check that H ( { z } , B ) = H ( G, M ) = Ω. Thus by[12, Proposition 2.23 and Proposition 2.25] the gauge function p B of B is arandom sublinear function[6] such that p B ( z ) > p B ( x ) < x ∈ B ◦ . Here B ◦ is the T c interior of B .Define g : { ξz | ξ ∈ L ( F , R ) } → L ( F , R ) by g ( ξz ) = ξp B ( z ), ∀ ξ ∈ L ( F , R ). It is easy to verify that g ( ξz ) p B ( ξz ), ∀ ξ ∈ L ( F , R ). Thus bythe Hahn-Banach extension theorem in RN module[1, Theorem2.8], thereexists a random linear functional f : E → L ( F , R ) such that f extends g and f ( x ) p B ( x ), ∀ x ∈ E .Notice that f ( x ) p B ( x ) < f ( x ) = − f ( − x ) > − p B ( x ) > − x ∈ B ◦ . It follows that f ( B ◦ ) ⊂ { ξ ∈ L ( F , R ) | | ξ | < } . Hence f is a T c continuous module homomorphism, i.e. f ∈ E ∗ . Moreover,since z − ( y − x ) ∈ B for any x ∈ G and y ∈ M , thus f ( x − y ) = f ( z − ( y − x )) − f ( z ) − p B ( z ) , i.e. f ( x ) f ( y ). Likewise, since z − ( y − x ) ∈ B ◦ for any x ∈ G ◦ and y ∈ M ,thus f ( x − y ) = f ( z − ( y − x )) − f ( z ) < − p B ( z ) , i.e. f ( x ) < f ( y ) on Ω. Hence f is the required random functional.If H ( G, M ) = Ω, let Ω ′ = H ( G, M ), F ′ = Ω ′ ∩ F = { Ω ′ ∩ F | F ∈ F } and P ′ : F ′ → [0 ,
1] be defined by P ′ (Ω ′ ∩ F ) = P (Ω ′ ∩ F ) /P (Ω ′ ). Take E ′ = ˜ I Ω ′ E , M ′ = ˜ I Ω ′ M , G ′ = ˜ I Ω ′ G and consider ( E ′ , k · k E ′ ) as an RN module with base (Ω ′ , F ′ , P ′ ). Then M ′ and G ′ satisfy the above condition,so there exists an f ′ ∈ ( E ′ ) ∗ such that6 ′ ( x ) f ′ ( y ) on Ω ′ for all x ∈ G ′ and y ∈ M ′ and f ′ ( x ) < f ′ ( y ) on Ω ′ for all x ∈ ( G ′ ) ◦ and y ∈ M ′ .By [1, Theorem 2.10] f ′ has an extension f ∈ E ∗ , which meets our require-ment.Finally, if R is replaced by C , the result follows immediately by noticingthat every RN module over C is also an RN module over R and f ( x ) = ( Ref )( x ) − i ( Ref )( ix ), ∀ f ∈ E ∗ and ∀ x ∈ E . ✷ Theorem 3.2.
Suppose ( E, k·k ) is an RN module over C with base (Ω , F , P ) and E has the countable concatenation property. f , f , · · · , f n ∈ E ∗ , ξ , ξ , · · · , ξ n ∈ L ( F , C ) and β ∈ L . For any ε ∈ L , there exists x ε ∈ E which satisfies:(1) f i ( x ε ) = ξ i , i = 1 , , · · · , n ;(2) k x ε k β + ε iff | n X k =1 λ k ξ k | β k n X k =1 λ k f k k holds for arbitrary λ , λ , · · · λ n ∈ L ( F , C ) . Proof.
Necessity is obvious, it remains to prove sufficiency.Let S = { Σ ni =1 ζ i f i | ζ i ∈ L ( F , C ) , i n } , then S is a finitelygenerated L ( F , C ) modules. By [14, Theorem 3.1], there exists a partition { A , A , · · · , A n } of Ω to F such that ˜ I A i S is a quasi-free stratification ofrank i of S for each i which satisfies 0 i n and P ( A i ) >
0. Let { g j ∈ ˜ I A i S | j i } be a basis for ˜ I A i S for some i such that 1 i n and P ( A i ) >
0. Suppose g j = Σ nk =1 ζ kj f k , 1 j i and ζ kj ∈ ˜ I A i L ( F , C ).Let γ j = Σ nk =1 ζ kj ξ k , then | i X j =1 λ j γ j | = | i X j =1 n X k =1 λ j ζ kj ξ k | β k i X j =1 n X k =1 λ j ζ kj f k k = β k i X j =1 λ j g j k . If there exists x A i ∈ ˜ I A i E such that g j ( x A i ) = γ j for 1 j i , then f k ( x A i ) = ˜ I A i ξ k for 1 k n . Actually, suppose ˜ I A i f k = P ij =1 η jk g j for7 k n and η jk ∈ ˜ I A i L ( F , C ), then | ˜ I A i ξ k − i X j =1 η jk γ j | β k ˜ I A i f k − i X j =1 η jk g j k = 0 , i.e. ˜ I A i ξ k = P ij =1 η jk γ j . Hence f k ( x A i ) = i X j =1 η jk g j ( x A i ) = i X j =1 η jk γ j = ˜ I A i ξ k . Thus no lose of generality, suppose { f , f , · · · , f n } is L ( F , C )-independent.Define T : E → L ( F , C n ) by T x = ( f ( x ) , f ( x ) , · · · , f n ( x )), ∀ x ∈ E .Obviously T ( E ) is a submodule of L ( F , C n ) with the countable concate-nation property. If T ( E ) = L ( F , C n ), by [14, Corollary 4.3] there exists z = ( η , η , · · · , η n ) ∈ L ( F , C n ) such that( n X k =1 ¯ η k f k )( x ) = n X k =1 ¯ η k f k ( x ) = h T ( x ) , z i = 0 , ∀ x ∈ E. This contradicts with the L ( F , C )-independence of { f , f , · · · , f n } . Hence T ( E ) = L ( F , C n ).Suppose x , x , · · · , x n ∈ E such that T x i = ( η i , η i , · · · , η in ), η ii = 1 and η ij = 0( i = j ) for 1 j n , 1 i n . Let γ = ∨ ni =1 k x i k , then clearly γ > y = ( α , α , · · · , α n ) ∈ L ( F , C n ) and k y k ( β + ε ) n − γ − for somefixed ε ∈ L , then T ( P ni =1 α i x i ) = y and k n X i =1 α i x i k n X i =1 | α i |k x i k ( β + ε ) n − γ − n X i =1 k x i k β + ε. Let ¯ B β + ε = { x ∈ E | k x k β + ε } , the above argument shows that T ( ¯ B β + ε )contains a T c -open neighborhood { y ∈ L ( F , C n ) | k y k < ( β + ε ) n − γ − } of the null element of L ( F , C n ). It is easy to see that T ( ¯ B β + ε ) is also an L -convex subset with the countable concatenation property.If the hypothesis does not hold, i.e. there exists ε ∈ L such that p , ( ξ , ξ , · · · , ξ n ) / ∈ T ( ¯ B β + ε ). Thus by Lemma 3.1 there exists f ∈ L ( F , C n ) ∗ such that ( Ref )( y ) ( Ref )( p ) on H ( { p } , T ( ¯ B β + ε )) for all y ∈ T ( ¯ B β + ε ) and8 Ref )( p ) > ( Ref )(0) = 0 on H ( { p } , T ( ¯ B β + ε )). For any fixed y ∈ T ( ¯ B β + ε ),let ξ = | f ( y ) | ( f ( y )) − , then clearly ξy ∈ T ( ¯ B β + ε ). Moreover, | f ( y ) | = f ( ξy ) = ( Ref )( ξy ) ( Ref )( p ) | f ( p ) | on H ( { p } , T ( ¯ B β + ε )).By Riesz’s representation theorem in RIP module [1, Theorem 4.3],there exists y = ( λ , λ , · · · λ n ) ∈ L ( F , C n ) such that f ( y ) = h y, y i , ∀ y ∈ L ( F , C n ). Hence | n X k =1 ¯ λ k f k ( x ) | = | f ( T x ) | | f ( p ) | = | n X k =1 ¯ λ k ξ k | on H ( { p } , T ( ¯ B β + ε )), ∀ x ∈ ¯ B β + ε . Thus( β + ε ) k n X k =1 ¯ λ k f k k = _ x ∈ ¯ B β + ε | n X k =1 ¯ λ k f k ( x ) | | n X k =1 ¯ λ k ξ k | . on H ( { p } , T ( ¯ B β + ε )).Since | f ( p ) | = |h p, y i| > H ( { p } , T ( ¯ B β + ε )), it follows k y k > H ( { p } , T ( ¯ B β + ε )). Thus k P nk =1 ¯ λ k f k k 6 = 0 on H ( { p } , T ( ¯ B β + ε )) by the L ( F , C )-independence of { f , f , · · · , f n } . Hence β k n X k =1 ¯ λ k f k k < | n X k =1 ¯ λ k ξ k | on H ( { p } , T ( ¯ B β + ε )), which contradicts with the assumption. ✷ Remark 3.3.
It is necessary to require E to have the countable concate-nation property in Theorem 4.2, otherwise the result may not hold. Hereis an example. Let Ω = [0 , F be the collection of all Lebesgue measur-able subsets of [0 ,
1] and P the Lebesgue measure on [0 , M = { ˜ I [2 − ( n +1) , − n ] | n ∈ N } and E = { P ni =1 ξ i x i | ξ i ∈ L ( F , C ) , x i ∈ M, i n and n ∈ N } . Clearly E is a submodule of L ( F , C ) without the countableconcatenation property. Define k · k : E → L by k η k = | η | , ∀ η ∈ E , then( E, k · k ) is also an RN module over C with base (Ω , F , P ). If f ∈ E ∗ is defined by f ( η ) = η , ∀ η ∈ E and take ξ = ˜ I Ω , β = ˜ I Ω . Then clearly | λξ | β k λf k (in fact, | λξ | = β k λf k ), ∀ λ ∈ L ( F , C ). But there does notexist any x ∈ E such that f ( x ) = ξ . 9 . The Goldstine-Weston theorem in RM modulesLemma 4.1. Let ( E, k · k ) be an RN module over C with base (Ω , F , P ) . If E has the countable concatenation property, then the unit ball E ∗ (1) = { f ∈ E ∗ | k f k } of the random conjugate space E ∗ of E is closed with respectto both σ ( ε,λ ) ( E ∗ , E ) and σ c ( E ∗ , E ) . Proof.
Since σ c ( E ∗ , E ) is stronger than σ ( ε,λ ) ( E ∗ , E ), it only needs to prove E ∗ (1) is σ ( ε,λ ) ( E ∗ , E ) closed. Suppose g ∈ E ∗ and g / ∈ E ∗ (1), then there exist A ∈ F and a positive real number δ such that P ( A ) > k g k > δ on A . By Theorem 3.2 there is an x ∈ E such that k x k (1 + δ/
2) ˜ I A and g ( x ) = ˜ I A k g k . Then the σ ( ε,λ ) ( E ∗ , E ) neighborhood N g ( x, δ/ , P ( A ) /
2) of g is disjoint with E ∗ (1), which completes the proof. ✷ Theorem 4.2.
Let ( E, k · k ) be an RN module over C with base (Ω , F , P ) with the countable concatenation property, J denote the random natural em-bedding E → E ∗∗ which is defined by J ( x )( g ) = g ( x ) for any g ∈ E ∗ and any x ∈ E . Then the σ c ( E ∗ , E ) closure of J ( E ) is E ∗∗ . Proof.
The result will be established immediately once we proved that the σ c ( E ∗ , E ) closure of J ( E (1)) is E ∗∗ (1). Suppose l ∈ E ∗∗ (1), f , f , · · · , f n ∈ E ∗ and ε ∈ L . Let γ = W ni =1 k f i k ∨ ˜ I Ω . By Theorem 3.2, there exists x ∈ E such that k x k ε k γ k − and f i ( x ) = l ( f i ) for 1 i n . Let x = γ ( γ + ε/ − x , then k x k | ( J ( x )( f i ) − l ( f i )) | = | f i ( x − x ) | ε k x k / < ε for 1 i n . Thus J ( x ) belongs to the σ c ( E ∗ , E ) neighborhood { h ∈ E ∗∗ | | ( h − l )( f i ) | < ε, i = 1 , , · · · , n } . Hence J ( E (1)) is σ c ( E ∗ , E ) dense in E ∗∗ (1), which completes the proof. ✷ Theorem 4.3.
Suppose ( E, k·k ) is an RN module over C with base (Ω , F , P ) ,then the σ ( ε,λ ) ( E ∗ , E ) closure of J ( E ) is E ∗∗ . Proof.
Suppose E cc = H cc ( E ) and define k · k cc : E cc → L by k x k cc = P n ∈ N ˜ I A n k x k n for any x = P n ∈ N ˜ I A n x n in E cc , where { A n | n ∈ N } is acountable partition of Ω to F and x n ∈ E for n ∈ N . It is easy to check that E ∗∗ cc = E ∗∗ . By the above theorem, J ( E cc ) is dense in E ∗∗ with respect to σ ( ε,λ ) ( E ∗ , E ) since σ c ( E ∗ , E ) is stronger than σ ( ε,λ ) ( E ∗ , E ). Combine the factthat J ( E ) is T ε,λ dense in J ( E cc ) with that T ε,λ is stronger than σ ( ε,λ ) ( E ∗ , E ),it follows our desired result. ✷ eferences [1] T.X. Guo, Relations between some basic results derived from two kindsof topologies for a random locally convex module, J. Funct. Anal.258(2010) 3024–3047.[2] T.X. Guo, Extension theorems of continuous random linear operatorson random domains, J. Math. Anal. Appl. 193(1)(1995) 15–27.[3] T.X. Guo, S.L. Peng, A characterization for an L ( µ, K )-topological mod-ule to admit enough canonical module homomorphisms, J. Math. Anal.Appl. 263(2001) 580–599.[4] T.X. Guo, Some basic theories of random normed linear spaces andrandom inner product spaces, Acta Anal. Funct. Appl. 1(2)(1999) 160–184.[5] T.X. Guo, The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-ˇSmulian theorem in complete random normed mod-ules to stratification structure, Sci China(Ser A)51(2008) 1651–1663.[6] T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem inrandom locally convex modules, Nonlinear Anal. 71(2009) 3794–3804.[7] T.X. Guo, L.H. Zhu, A characterization of continuous module homo-morphisms on random seminormed modules and its applications, ActaMath. Sinica English Ser. 19(1)(2003) 201–208.[8] T.X. Guo, Representation theorems of the dual of Lebesgue-Bochnerfunction spaces, Sci. China Ser. A-Math. 43(2000) 234–243.[9] T.X. Guo, Several applications of the theory of random conjugate spacesto measurability problems, Sci. China Ser. A-Math. 50(2007) 737–747.[10] T.X. Guo, and S.B. Li, The James theorem in complete random normedmodules, J. Math. Anal. Appl. 308(2005) 257–265.[11] N. Dunford, J. T. Schwartz, Linear Operators(I), Interscience, NewYork, 1957.[12] D.Filipovi´c, M.Kupper, N.Vogelpoth, Separation and and duality in lo-cally L -convex modules, J. Funct. Anal. 256(2009) 3996–4029.1113] T.X. Guo, Recent progress in random metric theory and its applicationsto conditional risk measures, Arxive: 1006.0697.[14] T.X. Guo, G. Shi, The algebraic structure of finitely generated L ( F , K, K