# On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules

Authors: Guo Tiexin
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We first prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules by simultaneously considering the two kinds of topologies—the(ϵ,λ)-topology and the locally L0-convex topology for random normed modules. Then, for the future development of the theory of module homomorphisms on complete random inner product modules, we give a proof with better readability of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under two kinds of topologies. Finally, to connect module homomorphism between random normed modules with linear operators between ordinary normed spaces, we give a proof with better readability of the known result connecting random conjugate spaces with classical conjugate spaces, namely, Lq(S*)≅(Lp(S))', where p and q are a pair of Hölder conjugate numbers with 1≤p<+∞,S a random normed module, S* the random conjugate space of S,Lp(S)(Lq(S*)) the corresponding Lp (resp., Lq) space derived from S (resp., S*), and (Lp(S))' the ordinary conjugate space of Lp(S).


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## Body

## 1. Introduction

The theory of probabilistic metric spaces initiated by K. Menger and subsequently developed by Schweizer and Sklar begins the study of randomizing the traditional space theory of functional analysis, where the randomness of “distance” or “norm” is expressed by probability distribution functions; compare [1]. The original notions of random metric spaces and random normed spaces occur in the course of the development of probabilistic metric and normed spaces, whereas the random distance between two points in a random metric space or the random norm of a vector in a random normed space is described by nonnegative random variables on a probability space; compare [1]. Probabilistic normed spaces are often endowed with the (ϵ,λ)-topology and not locally convex in general; a serious obstacle to the deep development of probabilistic normed spaces is that the taditional theory of conjugate spaces does not universally apply to probabilistic normed spaces. Although the traditional theory of conjugate spaces does not universally apply to random normed spaces either, the additional measure-theoretic structure and the stronger geometric structure peculiar to a random normed space enable us to introduce the notion of an almost surely bounded random linear functional and establish its Hahn-Banach extension theorem, which leads to the idea of the theory of random conjugate spaces for random normed spaces; compare [2–4].The further development of the theory of random conjugate spaces motivates us to present the important notions of random normed modules, random inner product modules, and random locally convex modules; compare [3–5]. Independent of Schweizer, Sklar, and Guo’s work, in [6] Haydon et al. also introduced random normed modules as a tool for the study of ultrapowers of Lebesgue-Bochner function spaces. All the work before 2009 was carried out under the (ϵ,λ)-topology.In 2009, motivated by financial applications, in [7] Filipović et al.  independently presented random normed modules and first applied them to the study of conditional risk measures. In particular, they introduced another kind of topology, namely, the locally L0-convex topology, for random normed modules and random locally convex modules, and began the study of random convex analysis.Relations between some basic results derived from the(ϵ,λ)-topology and the locally L0-convex topology for a random locally convex module were further studied in [8]. Following [8], the advantage and disadvantage of the two kinds of topologies are gradually realized and the advantage of one can complement the disadvantage of the other, which also leads to a series of recent advances [9, 10] and in particular to a complete random convex analysis with applications to conditional risk measures [11, 12].Up to now, the results obtained in random metric theory are of space-theoretical nature, whereas the study of module homomorphisms between random normed modules has not been fully carried out. With the development of random metric theory, we unavoidably need a deep theory of module homomorphisms; this paper gives some basic theorems of continuous module homomorphisms. These basic theorems are known under the(ϵ,λ)-topology, but their proofs were given before the definitive notions of random normed and inner product modules were presented in [3] so that these proofs do not have a good readability; in this paper we give better proofs and further give the versions of these basic theorems under the locally L0-convex topology.The remainder of this paper is organized as follows. In Section2, we introduce some basic notions together with some simple facts subsequently used in this paper. In Section 3, we prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules endowed with the two kinds of topologies. In Section 4, we give a better proof of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under the two kinds of topologies for the future development of module homomorphisms between complete random inner product modules. Finally, Section 5 is devoted to a better proof of the known result connecting random conjugate spaces and ordinary conjugate spaces, namely, (Lp(S))′≅Lq(S*).


## 2. Preliminaries

Throughout this paper,K denotes the scalar field, namely, the field R of real numbers or the field C of complex numbers, (Ω,ℱ,μ) a σ-finite measure space, L0(ℱ,K) the algebra of equivalence classes of ℱ-measurable K-valued functions on (Ω,ℱ,μ),  L-0(ℱ) the set of equivalence classes of extended real-valued ℱ-measurable functions on (Ω,ℱ,μ) and L0(ℱ):=L0(ℱ,R).L
-
0
(
ℱ
) is partially ordered by ξ≤η if and only if ξ0(ω)≤η0(ω)  a.s., where ξ0 and η0 are arbitrarily chosen representatives of ξ and η in L-0(ℱ), respectively. It is well known from [13] that every subset H in L-0(ℱ) has a supremum and infimum, denoted by ∨H and ∧H, respectively, and there are countable subsets {an,n∈N} and {bn,n∈N} of H such that ∨H=∨n≥1an and ∧H=∧n≥1bn. Furthermore, if, in addition, H is upward directed or downward directed, then {an,n∈N} and {bn,n∈N} can be chosen as nondecreasing and nonincreasing, respectively. In particular, (L0(ℱ),≤) is conditionally complete, namely, every subset with an upper bound has a supremum.Following are the notation and terminology frequently used in this paper:L-+0(ℱ)={ξ∈L-0(ℱ)∣ξ≥0},L+0(ℱ)={ξ∈L0(ℱ)∣ξ≥0},L-++0(ℱ)={ξ∈L0(ℱ)∣ξ>0onΩ},where “ξ>0 on Ω” means that ξ0(ω)>0  a.s. for an arbitrarily chosen representative ξ0 of ξ.Definition 1 (see [3]).
An ordered pair(S,∥·∥) is called a random normed space (briefly, an RN space) over K with base (Ω,ℱ,μ) if S is a linear space over K and ∥·∥ is a mapping from S to L+0(ℱ) such that the following three conditions are satisfied:(RN-1)
∥x∥=0 implies x=θ (the null in S),(RN-2)
∥αx∥=|α|∥x∥, for all α∈K and x∈S,(RN-3)
∥x+y∥≤∥x∥+∥y∥, for all x,y∈S,where ∥x∥ is called the random norm of x. If ∥·∥ only satisfies (RN-2) and (RN-3), then it is called a random seminorm.Furthermore, if, in addition,S is a left module over the algebra L0(ℱ,K) (briefly, an L0(ℱ,K)-module) and the following additional condition is also satisfied:(RNM-1)
∥ξx∥=|ξ|∥x∥, for all ξ∈L0(ℱ,K) and x∈S,then (S,∥·∥) is called a random normed module (briefly, an RN module) over K with base (Ω,ℱ,μ), at which time ∥·∥ is called an L0 norm on S. Similarly, if ∥·∥ only satisfies (RN-3) and (RNM-1), then it is called an L0-seminorm on S.Remark 2.
In [1], the original definition of an RN space was given by only requiring (Ω,ℱ,μ) to be a probability space and defining ∥x∥ to be a nonnegative random variable; the corresponding (RN-1) to (RN-3) are given in the following way:(RN-1)′
∥x∥(ω)=0  a.s. implies x=θ,(RN-2)′
∥αx∥(ω)=|α|∥x∥(ω)a.s., for all α∈K and x∈S,(RN-3)′
∥x+y∥(ω)≤∥x∥(ω)+∥y∥(ω)a.s., for all x,y∈S.This definition is natural and intuitive from probability theory, but (RN-1)′ is difficult to satisfy when we construct examples. Thus we essentially have employed Definition1 since our work [2] by saying that measurable functions or random variables that are equal a.s. are identified; in particular since 1999 we strictly distinguish between measurable functions and their equivalence classes in writings; compare [3].Remark 3.
At outset we consider both the real and complex cases in the study of RN spaces, whereas they only consider the real case in [6, 7] because of their special interests; an RN module over R is termed as a randomly normed L0-module in [6] and an L0-normed module in [7]. We still would like to continue to employ the terminology “an RN module over K with base (Ω,ℱ,μ)” in order to keep concordance with the earliest terminology used in [1].Definition 4 (see [3, 5, 14]).
Let(S,∥·∥) and (S1,∥·∥1) be two RN spaces over K with base (Ω,ℱ,μ). A linear operator T from S to S1 is said to be a.s. bounded if there exists ξ∈L+0(ℱ) such that ∥Tx∥1≤ξ∥x∥,forallx∈S, in which case ∥T∥ is defined to be ∧{ξ∈L+0(ℱ)∣∥Tx∥1≤ξ∥x∥,∀x∈S}, called the random norm of T. Denote the linear space of a.s. bounded linear operators from S to S1 by B(S,S1); then (B(S,S1),∥·∥) still becomes an RN space over K with base (Ω,ℱ,μ) when ∥T∥ is defined as above for every T∈B(S,S1). In particular, when S1=L0(ℱ,K) and ∥·∥1=|·| (namely, the absolute value mapping), S*:=B(S,S1) is called the random conjugate space of S and an element in S* is called an a.s. bounded random linear functional on S.Remark 5.
When(S1,∥·∥1) in Definition 4 is an RN module, B(S,S1) automatically becomes an RN module under the module operation (ξ·T)(x)=ξ·(T(x)), for all ξ∈L0(ℱ,K),T∈B(S,S1), and x∈S. When S and S1 are both RN modules, in [6] B(S,S1) is used to stand for the L0(ℱ,K)-module of a.s. bounded module homomorphisms from S to S1; we will show that in the special case an a.s. bounded linear operator must be a module homomorphism. Therefore, the two implications of B(S,S1) coincide in this case.As in the classical functional analysis, we can similarly define a conjugate operatorT*:S2*→S1* for an a.s. bounded linear operator T from (S1,∥·∥1) to (S2,∥·∥2) as follows: (T*f)(x)=f(Tx), for all f∈S2*,x∈S1. From the Hahn-Banach theorem for a.s. bounded random linear functional established in [2] (also see [8]), one has that ∥T*∥=∥T∥.For the sake of convenience, let us recall some notation and terminology in the theory of probabilistic normed spaces.Definition 6 (see [1]).
A functionT:[0,1]×[0,1]→[0,1] is called a weak t-norm if the following are satisfied: (t-1)
T(a,b)=T(b,a), for all a,b∈[0,1],(t-2)
T(a,b)≤T(c,d), for all a,b,c,d∈[0,1] with a≤c,b≤d,(t-3)
T(1,0)=0,T(1,1)=1.A weak t-norm T:[0,1]×[0,1]→[0,1] is called a t-norm if the following two additional conditions are satisfied:(t-4)
T(a,T(b,c))=T(T(a,b),c), for all a,b,c∈[0,1],(t-5)
T(1,a)=a, for all a∈[0,1].Althought-norms are widely used in the theory of probabilistic metric spaces, weak t-norms have their own advantages, for example, for a family {Tα,α∈∧} of weak t-norms, T defined by T(a,b)=sup{Tα(a,b):α∈∧}, for all a,b∈[0,1], is still a weak t-norm, whereas this is not true for t-norms.Throughout this paper,Δ={F:[-∞,+∞]→[0,1]∣F(-∞)=0,F(+∞)=1,F is nondecreasing and left continuous on (-∞,+∞)}, D={F∈Δ∣limx→-∞F(x)=0 and limx→+∞F(x)=1},Δ+={F∈Δ∣F(0)=0}, and D+={F∈D∣F(0)=0}. For extended real random variable ξ on a probability space (Ω,ℱ,P), its (left continuous) distribution function Nξ is defined by Nξ(t)=P{ω∈Ω∣ξ(ω)<t}, for  all t∈[-∞,+∞)andNξ(+∞)=1.In particular,ϵ0 stands for the distribution function defined by ϵ0(t)=1 when t>0 and ϵ0(t)=0 when t≤0; namely, ϵ0 is the distribution function of the constant 0.Definition 7 (see [1]).
A triple(S,𝒩,T) is called a Menger probabilistic normed space (briefly, an M-PN space) over K if S is a linear space over K,𝒩 is a mapping from S to D+, and T is a weak t-norm such that the following are satisfied:(MPN-1)
Nx=ϵ0 if and only if x=θ(the null element in S),(MPN-2)
Nαx(t)=Nx(t/|α|), for all t≥0,α∈K∖{0} and x∈S,(MPN-3)
Nx+y(u+v)≥T(Nx(u),Ny(v)), for all u,v≥0 and x,y∈S. Here Nx:=𝒩(x) is called the probabilistic norm of x.For an M-PN space(S,𝒩,T), let 𝒯={T~∣T~ is a weak t-norm such that (S,𝒩,T~)isanM-PNspace}, and define T^:[0,1]×[0,1]→[0,1] by T^(a,b)=sup{T~(a,b)∣T~∈𝒯}, for all a,b∈[0,1]; then it is very easy to see that T^∈𝒯.T^ is called the largest weak t-norm of (S,𝒩) such that (S,𝒩) is an M-PN space under T^. From now on, for an M-PN space (S,𝒩,T), we always assume that T is the largest weak t-norm of (S,𝒩).In [15], LaSalle introduced the notion of a pseudonormed linear space: let S be a linear space over K and {pα}α∈∧ a family of mappings from S to R+:=[0,+∞) and indexed by a directed set ∧; then (S,{pα}α∈∧) is called a pseudonormed linear space if the following are satisfied:(PNS-1)
pα(βx)=|β|pα(x), for all β∈K,x∈S, and α∈∧,(PNS-2)
pα1(x)≤pα2(x), for all α1,α2∈∧ such that α1≤α2,x∈S,(PNS-3)
for anyα∈∧, there exists α′∈∧ such that pα(x+y)≤pα′(x)+pα′(y), for all x,y∈S.Let(S,{pα}α∈∧) be a pseudonormed linear space. For any ϵ>0 and α∈∧, let Uθ(α,ϵ)={x∈S∣pα(x)<ϵ}. Then 𝒰θ={Uθ(α,ϵ)∣α∈∧,ϵ>0} is a local base at the null element θ of some linear topology for S, called the linear topology induced by {pα}α∈∧. Conversely any linear topology for S can be induced by some {pα}α∈∧ such that (S,{pα}α∈∧) is a pseudonormed linear space.To connect an M-PN space(S,𝒩,T) to a pseudonormed linear space, for each r∈(0,1), define pr:S→[0,+∞) by pr(x)=sup{t≥0∣Nx(t)<r}, for all x∈S. Then we have the following.Theorem 8 (see [16]).
Let(S,𝒩,T) be an M-PN space. Then one has the following statements.(1)
sup0<a<1T(a,a)=1 if and only if (S,{pr}r∈(0,1)) is a pseudonormed linear space; namely, for each r∈(0,1) there exists s∈(0,1) such that pr(x+y)≤ps(x)+ps(y), for all x,y∈S.(2)
T≥Min, namely, T(a,b)≥min(a,b), for all a,b∈[0,1], if and only if pr is a seminorm on S for each r∈(0,1); namely, (S,{pr}r∈(0,1)) is a B0-type space.(3)
T(a,b)=1 for all a,b∈[0,1] such that a·b>0 if and only if there exists a norm ∥·∥ on S such that pr=∥·∥, for all r∈(0,1).Theorem8 was first studied in [17] in terms of isometric metrization and first given and strictly proved in its present form in [16].Proposition 9 (see [1]).
Let(S,𝒩,T) be an M-PN space such that sup0<a<1T(a,a)=1. For any positive numbers ϵ and λ with 0<λ<1, let Uθ(ϵ,λ)={x∈S∣Nx(ϵ)>1-λ}; then 𝒰θ={Uθ(ϵ,λ)∣ϵ>0,0<λ<1} forms a local base at θ of some metrizable linear topology for S, called the (ϵ,λ)-topology induced by 𝒩.From Theorem8, one can easily see that the (ϵ,λ)-topology for an M-PN space (S,𝒩,T) with sup0<a<1T(a,a)=1 is exactly the one induced by the family {pr}r∈(0,1) of pseudonorms. Therefore as far as the study of linear homeomorphic invariants for a metrizable linear topological space is concerned, the theory of an M-PN space (S,𝒩,T) with sup0<a<1T(a,a)=1 and the theory of pseudonormed linear space (S,{pr}r∈(0,1)) are equivalent to each other, and hence either of them is also equivalent to the theory of a quasinormed space (see [18] for a quasinormed space) since a metrizable linear topology can be equivalently induced by a quasinorm as well as a family of pseudonorms such as {pr}r∈(0,1). We find that the three kinds of frameworks have their own advantages and all will be used in this paper.Definition 10 (see [1]).
Let(S,𝒩,T) be an M-PN space with sup0<a<1T(a,a)=1 and A a subset of S.DA:[-∞,+∞]→[0,1] is defined by DA(t)=supr<t(inf{Nx(r):x∈A}), for all t∈(-∞,+∞),DA(-∞)=0, and DA(+∞)=1, called the probabilistic diameter of A. If DA∈D+, then A is said to be probabilistically bounded.Proposition11 below is a straightforward verification by definition.Proposition 11.
Let(S,𝒩,T) and A be the same as in Definition 10. Then A is probabilistically bounded if and only if A is bounded with respect to the (ϵ,λ)-topology (namely, A can be absorbed by every (ϵ,λ)-neighborhood of the null θ).Let(Ω,ℱ,μ) be a probability space and (S,∥·∥) an RN space over K with base (Ω,ℱ,μ). Define 𝒩:S→D+ by Nx(t)=μ{ω∈Ω∣∥x∥(ω)<t},forallt≥0 and x∈S; namely, Nx is the distribution of ∥x∥; then (S,𝒩,T) is an M-PN space with T≥W, where W(a,b)=max(a+b-1,0), for all a,b∈[0,1].(S,𝒩,T) is called the M-PN space determined by (S,∥·∥); the (ϵ,λ)-topology for (S,𝒩,T) is also called the (ϵ,λ)-topology for (S,∥·∥).When(Ω,ℱ,μ) is a σ-finite measure space, let ℱ+={A∈ℱ∣0<μ(A)<+∞}; then the following definition is a slight generalization of the case when (Ω,ℱ,μ) is a probability space.Definition 12 (see [3]).
Let(S,∥·∥) be an RN space over K with base (Ω,ℱ,μ). For A∈ℱ+,ϵ>0 and λ>0, let Uθ(A,ϵ,λ)={x∈S∣μ{ω∈A∣∥x∥(ω)<ϵ}>μ(A)-λ}. Then 𝒰θ={Uθ(A,ϵ,λ)A∈ℱ+(A),ϵ>0,λ>0} forms a local base at θ of some metrizable linear topology for S, called the (ϵ,λ)-topology for S induced by ∥·∥.Proposition13 below is a straightforward verification by definition.Proposition 13.
Let(S,∥·∥) be an RN space over K with base (Ω,ℱ,μ) and {An∣n∈N} a countable partition of Ω to ℱ such that 0<μ(An)<+∞. Then one has the following.(1)
A sequence{xnx∈N}inSconverges in the(ϵ,λ)-topology toxinSif and only if{∥xn-x∥∣n∈N}converges to
0
locally in measure; namely,{∥xn-x∥∣n∈N}converges to
0
in measureμon eachA∈ℱ+.(2)
The(ϵ,λ)-topology forSis exactly the linear topology induced by the quasinorm|∥·∥|defined by|∥x∥|=∑n=1∞‍(1/2n)∫An‍(∥x∥)/(1+∥x∥)dμfor allx∈S.(3)
LetP:ℱ→[0,1]be defined byP(A)=∑n=1∞‍(1/2n)(μ(A∩An)/μ(An)); thenPis a probability measure equivalent toμand(S,∥·∥)has the same(ϵ,λ)-topology whether(S,∥·∥)is regarded as an RN space with base(Ω,ℱ,μ)or(Ω,ℱ,P).Remark 14.
When(Ω,ℱ,μ) is a σ-finite measure space, the (ϵ,λ)-topology for the special RN space (L0(ℱ),|·|) is exactly the topology of convergence locally in measure. But the topology of convergence in measure is not a linear topology in general, so we choose the (ϵ,λ)-topology since not only is it a linear topology but also its convergence has almost all the nice properties of convergence in measure (see (1) of Proposition 13). (3) of Proposition 13 shows that we can always assume the base space of an RN space to be a probability space when only the linear homeomorphic invariants or those independent of the special choice of μ and P are studied. Finally, independently of B Schweizer and Sklar’s work [1], the (ϵ,λ)-topology is also introduced in [6], called the L0-topology.Definition 15 (see [3, 5, 14]).
Let(S,∥·∥) be an RN space over K with base (Ω,ℱ,μ) and A a subset of S.A is said to be a.s. bounded if ∨{∥a∥:a∈A}∈L+0(ℱ).In the sequel, the(ϵ,λ)-topology for every RN space is always denoted by 𝒯ϵ,λ and the quasinorm for every RN space is always denoted by |∥·∥| defined as in (2) of Proposition 13 when no confusion occurs.Proposition 16 (see [3]).
Let(S,∥·∥) be an RN space with base (Ω,ℱ,μ) and A a subset of S such that {∥a∥:a∈A} is upward directed. Then A is a.s. bounded if and only if it is 𝒯ϵ,λ-bounded, at which time and when (Ω,ℱ,μ) is a probability space, DA=Nξ, where ξ=∨{∥a∥:a∈A} and Nξ is the distribution function of ξ.Proof.
We can, without loss of generality, assume that(Ω,ℱ,μ) is a probability space. Necessity is clear. We prove the sufficiency as follows.
Since there exists a sequence{xn,n∈N} in A such that {∥xn∥,n∈N} converges a.s. to ξ in a nondecreasing manner. Let (S,𝒩,T) be the M-PN space determined by (S,∥·∥); then {Nxn,n∈N} converges weakly to Nξ; it is easy to check that Nξ=D{xn,n∈N} (namely, the probabilistic diameter of {xn,n∈N}), and hence Nξ≥DA. But Nξ≤DA is clear, then Nξ=DA. Since A is 𝒯ξ,λ-bounded, DA∈D+, which shows that ξ∈L+0(ℱ).Proposition17 below gives a very general condition for {∥a∥:a∈A} to be upward directed or downward directed.Proposition 17.
Let(S,∥·∥) be an RN module with base (Ω,ℱ,μ) and A a subset of S such that I~DA+I~DCA⊂A for any D∈ℱ, where DC=Ω∖D and I~D stands for the equivalence class of the characteristic function of D. Then {∥a∥:a∈A} is both upward and downward directed.Proof.
We only prove that{∥a∥:a∈A} is upward directed; the case of being downward directed is similar. For any a1,a2∈A, let D={ω∈Ω∣∥a1∥0(ω)≤∥a2∥0(ω)}, where ∥a1∥0 and ∥a2∥0 are arbitrarily chosen representatives of ∥a1∥ and ∥a2∥, respectively. Then a3∶=I~Da2+I~DCa1 is such that ∥a3∥=I~D∥a2∥+I~DC∥a1∥=∥a1∥∨∥a2∥. Since a3∈A, the proof is complete.It is easy to see that(L0(ℱ,K),𝒯ϵ,λ) is a topological algebra over K and (S,𝒯ϵ,λ) is a topological module over (L0(ℱ,K),𝒯ϵ,λ) when (S,∥·∥) is an RN module over K with base (Ω,ℱ,μ). In 2009, another kind of topology for an RN module was introduced in [7].Definition 18 (see [7]).
Let(S,∥·∥) be an RN module over K with base (Ω,ℱ,μ). A subset G of S is called a 𝒯c-open set if for each x∈G there exists some ϵ∈L++0(ℱ) such that x+Uθ(ϵ)⊂G, where Uθ={y∈S∣∥y∥≤ϵ}. Denote by 𝒯c the family of 𝒯c-open sets; then 𝒯c forms a Hausdorff topology for S, called the locally L0-convex topology induced by ∥·∥.It is easy to check that the locallyL0-convex topology is much stronger than the (ϵ,λ)-topology for a given RN module; (L0(ℱ,K),𝒯c) is, however, only a topological ring since it is unnecessarily a linear topological space (see [7]). Furthermore, for an RN module (S,∥·∥) over K with base (Ω,ℱ,μ),(S,𝒯c) is a topological module over the topological ring (L0(ℱ,K),𝒯c);compare [7]. From now on, the locally L0-convex topology for every RN module is always denoted by 𝒯c when no confusion occurs.Definition 19.
LetS be an L0(ℱ,K)-module. A subset G of S is said to be L0-convex if ξx+(1-ξ)y∈G, for all x,y∈G and ξ∈L+0(ℱ) with 0≤ξ≤1. A subset G of S is said to be L0-balanced if ξx∈G for all x∈G and ξ∈L0(ℱ,K) with |ξ|≤1. A subset G of S is said to be L0-absorbed by a subset H of S if there exists some ξ∈L++0(ℱ) such that ηG⊂H for all η∈L0(ℱ,K) with |η|≤ξ. Furthermore, if a subset G of SL0-absorbs every element of S, then G is said to be L0-absorbent.Definition 20 (see [12]).
Let(S,∥·∥) be an RN module and A a subset of S.A is said to be 𝒯c-bounded if A is L0-absorbed by every 𝒯c-neighborhood of the null element.It is also very easy to see that a subset of an RN module is𝒯c-bounded if and only if it is a.s. bounded.For the sake of convenience,IA always denotes the characteristic function of A∈ℱ and I~A the equivalence class of IA. As usual, {B∈ℱ∣μ(AΔB)=0} is called the equivalence class of A∈ℱ, denoted by A~; we sometimes also use IA~ for I~A.Theorem21 below is a formal generalization of the corresponding results given in [5, 19] for a random linear functional; it was already frequently employed in [12, 14] but does not have yet a better proof; here we give a better proof. From now on, for convenience we always denote by (S,N) the M-PN space determined by a given RN space (S,∥·∥).Theorem 21.
Let(S1,∥·∥1) and (S2,∥·∥2) be two RN modules over K with base (Ω,ℱ,μ) and T:S1→S2 a linear operator. Then one has the following: (1)
T is a.s. bounded if and only if T is a continuous module homomorphism from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ);(2)
T is a.s. bounded if and only if  T is a continuous module homomorphism from (S1,𝒯c) to (S2,𝒯c).Proof(1) Necessity. Since T is a.s. bounded, T must be continuous from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ); it remains to prove that T is also a module homomorphism; it suffices to prove that T(I~Ax)=I~AT(x), for all x∈S1 and A∈ℱ, since T is linear. Since ∥T(I~Ax)∥2≤∥T∥·∥I~Ax∥1=I~A∥T∥·∥x∥1, I~Ac·T(I~Ax)=θ, for all A∈ℱ. Then T(I~Ax)=(I~A+I~Ac)·T(I~Ax)=I~A·T(I~Ax), for all A∈ℱ. On the other hand, I~AT(x)=I~A·T(I~Ax+I~Acx)=I~A·T(I~Ax), for all x∈S1 and for all A∈ℱ. So, I~A·T(x)=T(I~Ax).Sufficiency. S1(1):={x∈S1∣∥x∥1≤1} is a.s. bounded, and hence also 𝒯ϵ,λ-bounded; further T(S1(1)) is 𝒯ϵ,λ-bounded since T is a continuous linear operator. Besides, I~A·T(S1(1))+I~Ac·T(S1(1))⊂T(S1(1)) for all A∈ℱ since S1(1) has this property and T is a module homomorphism. Then T(S1(1)) is a.s. bounded; namely, ξ:=∨{∥T(x)∥2:x∈S1(1)}∈L+0(ℱ) by Propositions 16 and 17. Since 1/(∥x∥1+(1/n))·x∈S1(1), for all x∈S1 and n∈N, ∥T(1/(∥x∥1+(1/n))·x)∥2≤ξ, which implies that ∥T(x)∥2≤ξ·∥x∥1, for all x∈S1; that is to say, T is a.s. bounded, at which time it is also clear that ∥T∥=∨{∥T(x)∥2:x∈S1(1)}.(2) Necessity. From the proof of necessity of (1), if T is a.s. bounded then T is a module homomorphism. The fact that T is a.s. bounded also obviously implies that T is continuous from (S1,𝒯c) to (S2,𝒯c).Sufficiency. Since S2(1):={y∈S2∣∥y∥≤1} is a 𝒯c-neighborhood of the null element of S2 there exists some ϵ∈L++0(ℱ) such that T(Uθ(ϵ))⊂S2(1), where Uθ(ϵ)={x∈S1∣∥x∥1≤ϵ}. Thus for any x∈S1, ∥T(1/(∥x∥1+(1/n))·x)∥2≤1, for all x∈S1, and n∈N; namely, ∥T(x)∥2≤(1/ϵ)(∥x∥1+(1/n)) by the fact that T is a module homomorphism, which shows that ∥T(x)∥2≤(1/ϵ)∥x∥1, for all x∈S1; namely, T is a.s. bounded.Remark 22.
(
1
) of Theorem 21 was independently obtained by Guo in [5] and Haydon et al. in [6] although it is stated in a different way in [6, Proposition5.6], one careful reader can see that Proposition5.6 of [6] exactly amounts to (1) of Theorem 21. Our proof is different from Haydon et al.’s in that we make use of something from the theory of Menger-PN spaces (see the proof of Proposition 16) and our method may also be used in the proofs of some important results of Section 3.Remark 23.
Let(S1,∥·∥1) and (S2,∥·∥2) be two RN modules over K with base (Ω,ℱ,μ); when T is a continuous module homomorphism from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ) or from (S1,𝒯c) to (S2,𝒯c), the process of proof of Theorem 21 has implied that ∥T∥=∨{∥T(x)∥2∣x∈S1(1)}; further we have NT=DT(S1(1)) by Proposition 16, where NT is the probabilistic norm of T, namely, the distribution function of ∥T∥, and DT(S1(1)) is the probabilistic diameter of T(S1(1)).The proof of Proposition24 below is similar to that of (1) of Theorem 21, so is omitted, but this proposition is very useful in the proof of the resonance theorem in Section 3 of this paper; we state it as follows.Proposition 24 (see [14]).
Let(S,∥·∥) be an RN module over K with base (Ω,ℱ,μ) and f:S→L+0(ℱ) such that the following two conditions are satisfied: (1)
f(αx)=α·f(x), for allx∈Sand all nonnegative numbersα;(2)
f(x+y)≤f(x)+f(y),
for allx,y∈S.Then f is a.s. bounded; namely, there is some ξ∈L+0(ℱ) such that f(x)≤ξ∥x∥, for allx∈S, if and only if f is continuous from (S,𝒯ϵ,λ) to (L0(ℱ),𝒯ϵ,λ) and f(ηx)=η·f(x),forallx∈S and η∈L+0(ℱ), at which time ∥f∥=∨{f(x)∣x∈S(1)}, where ∥f∥=∧{ξ∈L+0(ℱ)∣f(x)≤ξ∥x∥,∀x∈S}; furthermore if, in addition, (Ω,ℱ,μ) is a probability space, then Nf (the distribution function of ∥f∥)=Df(S(1)).It is well known thatB(S1,S2) is a Banach space when S1 and S2 are normed spaces and S2 is complete. Similarly, B(S1,S2) is 𝒯ϵ,λ-complete when (S1,∥·∥1) and (S2,∥·∥2) are RN modules and S2 is 𝒯ϵ,λ-complete, which is independently pointed out by Guo in [5, 14] and Haydon et al. in [6]; in particular S* is 𝒯ϵ,λ-complete for every RN module S. In fact, a more general result is proved in [14], namely, the following.Proposition 25 (see [14]).
Let(S1,∥·∥1) and (S2,∥·∥2) be two RN spaces over K with base (Ω,ℱ,μ) such that S2 is 𝒯ϵ,λ-complete; then B(S1,S2) is 𝒯ϵ,λ-complete.WhenS1 and S2 are both RN modules, since ∥T∥=∨{∥T(x)∥2∣x∈S1(1)}, for any T∈B(S1,B2), the proof of Proposition 25 is similar to that of the classical case. But when S1 and S2 are only RN spaces, its proof needs Lemma 26 below. To state it, let us recall the canonical embedding mapping J from an RN space (S,∥·∥) to (S**,∥·∥**), where S**=(S*)*,J(x) is defined by (J(x))(f)=f(x), for all f∈S* and x∈S. It is easy to see that J is random-norm preserving. As usual, S is said to be random reflexive if J is surjective. Generally, the 𝒯ϵ,λ-closed submodule generated by J(S) in S** is called the 𝒯ϵ,λ-closed submodule generated by S, denoted by M(S); it is, obviously, a 𝒯ϵ,λ-complete RN module.Lemma26 below is given and proved in [14]; here we give it a better proof.Lemma 26 (see [14]).
Let(S1,∥·∥1) and (S2,∥·∥2) be two RN spaces over K with base (Ω,ℱ,μ) such that S2 is 𝒯ϵ,λ-complete. Then B(S1,B2) is isomorphic to a 𝒯ϵ,λ-closed subspace of B(M(S1,M(S2)) in a random-norm-preserving way.Proof.
DefineL:B(S1·S2)→B(M(S1),M(S2)) by L(T)=T**∣M(S1), where T** is the random conjugate operator of T*.
First,L is well defined, namely, L(T)(M(S1))⊂M(S2), and isometric. Let J1:S1→S1** and J2:S2→S2** be the corresponding canonical embedding mappings; it is easy to check that T**∘J1=J2∘T, which not only shows that T**(J1(S1))⊂J2(S2) but also shows that ∥T∥=∥T**∣J1(S1)∥≤∥L(T)∥≤∥T**∥. Since ∥T∥=∥T**∥,∥L(T)∥=∥T∥. Further by (1) of Theorem 21 one can easily see that L(T)(M(S1))⊂M(S2).
Second,L(B(S1,S2)) is a 𝒯ϵ,λ-closed subspace of B(M(S1),M(S2)). Let {Tn,n∈N} be a sequence in B(S1,S2) such that {L(Tn),n∈N} converges in the (ϵ,λ)-topology to some T-∈B(M(S1),M(S2)). Then {Tn,n∈N} is also 𝒯ϵ,λ-Cauchy in B(S1,S2). We can, without loss of generality, assume that {∥Tn∥,n∈N} converges a.s. to some ξ. Define T:S1→S2 by T(x)=𝒯ϵ,λ-limn→∞Tn(x),forallx∈S1; then T is well defined since S2 is 𝒯ϵ,λ-complete, and T is a.s. bounded since ∥T(x)∥2≤ξ∥x∥, for all x∈S1. Finally, it is easy to check that L(T)=T-.Remark 27.
SinceB(M(S1),M(S2)) is always 𝒯ϵ,λ-complete, so is B(S1,S2) when S2 is 𝒯ϵ,λ-complete by Lemma 26.


## 3. Some Basic Principles of Continuous Module Homomorphisms between Random Normed Modules

The main purpose of this section is to generalize some classical basic principles such as the resonance theorem, open mapping theorem, closed graph theorem, and inverse operator theorem to the context of random normed modules. It turns out that the counterparts under the(ϵ,λ)-topology are consequences of the corresponding classical theorems on ordinary operators between quasinormed spaces except for the proof of the resonance theorem which is somewhat complicated. However, the counterparts under the locally L0-convex topology are another thing since the usual reasoning fails to be valid; for example, the Baire category argument is no longer valid. Owing to the relations established in [8], we can prove them by converting their proofs to the case for the (ϵ,λ)-topology.The following surprisingly general uniform boundedness theorem is known (see [18]). But for the sake of reader’s convenience, we state it as follows.Proposition 28 (see [18]).
LetX be a linear topological space over K of second category and (Y,|∥·∥|) a quasinormed linear space. Let {Tα,α∈∧} be a family of continuous mappings from X to Y such that the following three properties are satisfied:(1)
|∥Tα(x+y)∥|≤|∥Tα(x)∥|+|∥Tα(y)∥|,
for allx,y∈Xandα∈∧;(2)
|∥Tα(ax)∥|=|∥aTα(x)∥|,
for allx∈X,α∈∧, anda≥0;(3)
{Tα(x),α∈∧}is bounded with respect to the linear topology induced by|∥·∥|for eachx∈X. Then limx→θTα(x)=θ uniformly in α∈∧.Theorem 29.
Let(S1,∥·∥1) and (S2,∥·∥1) be two RN modules over K with base (Ω,ℱ,μ) such that S1 is 𝒯ϵ,λ-complete. Let {Tα:α∈∧}. be a family of continuous module homomorphisms from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ). Then, one has the following: (1)
{Tα:α∈∧} is 𝒯ϵ,λ-bounded in B(S1,S2) if and only if {Tα(x):α∈∧} is 𝒯ϵ,λ-bounded in S2 for each x∈S1;(2)
{Tα:α∈∧} is a.s. bounded in B(S1,S2) if and only if {Tα(x):α∈∧} is a.s. bounded in S2 for each x∈S1.Proof.
We can, without loss of generality, assume that(Ω,ℱ,μ) is a probability space.(1)  Necessity. Let {Tα:α∈∧} be 𝒯ϵ,λ-bounded in B(S1,S2); namely, D{Tα:α∈∧}∈D+. For each x∈S1, since ∥Tα(x)∥2≤∥Tα∥∥x∥1,forallα∈∧, μ{ω∈Ω∣∥Tα(x)∥2(ω)<t}≥μ({ω∈Ω∣∥Tα∥2(ω)<2t}∩{ω∈Ω∣∥x∥1(ω)<(1/2)t})≥μ{ω∈Ω∣∥Tα∥2(ω)<2t}+μ{ω∈Ω∣∥x∥1(ω)<(1/2)t}-1; namely, NTα(x)(t)≥NTα(2t)+Nx((1/2)t)-1, for all α∈∧andt>0. Then, D{Tα(x):α∈∧}(t)≥W(D{Tα:α∈∧}(2t),Nx((1/2)t)), for all x∈S1 and t>0, where W(a,b)=max(a+b-1,0), for all a,b∈[0,1]. Since D{Tα:α∈∧}∈D+andNx∈D+ for any x∈S1, then D{Tα(x):α∈∧}∈D+; namely, {Tα(x):α∈∧} is 𝒯ϵ,λ-bounded in S2 for each x∈S1.Sufficiency. Let |∥·∥|:S2→[0,+∞) be defined by |∥y∥|=∫Ω‍(∥y∥/(1+∥y∥))dμ for any y∈S2; then (S2,|∥·∥|) is a quasinormed linear space and |∥·∥| induces the (ϵ,λ)-topology for S2. Since (S1,𝒯ϵ,λ) is a linear topological space of the second category and it is also clear that {Tα:α∈∧} satisfies all the three conditions of Proposition 28, limx→θTα(x)=0 uniformly in α∈∧ by Proposition 28, which certainly implies that ⋃α∈∧Tα(A) is 𝒯ϵ,λ-bounded in S2 for each 𝒯ϵ,λ-bounded set A in S1, in particular ⋃α∈∧Tα(S1(1)) is 𝒯ϵ,λ-bounded. By Remark 23, NTα=DTα(S1(1)), for all α∈∧. Define F:[-∞,+∞]→[0,1] by F(t)=infα∈∧DTα(S1(1))(t), for all t∈[-∞,+∞], and l-F:[-∞,+∞] by (l-F)(-∞)=0,(l-F)(+∞)=1, and (l-F)(t)=sup{F(t′)∣t′<t}, forallt∈(-∞,+∞); denote ⋃α∈∧Tα(S1(1)) by A; then one can easily check that DA=l-F=D{Tα:α∈∧}; then {Tα:α∈∧} is 𝒯ϵ,λ-bounded since DA∈D+.(2) Necessity of (2) Is Clear. We prove sufficiency of (2) as follows.Denote the family of finite subsets of∧ by ℱ(∧). For any F∈ℱ(∧), define fF:S1→L+0(ℱ) by fF(x)=∨{∥Tαx∥2∣α∈F}, for all x∈S1; then fF is continuous from (S1,𝒯ϵ,λ) to (L0(ℱ,K),𝒯ϵ,λ) and fF(ξx)=ξ·fF(x), for all ξ∈L+0(ℱ) and x∈S1, and hence fF is a.s. bounded and ∥fF∥=∨{fF(x)∣x∈S1(1)}=∨{∥Tα∥:α∈F}. It is obvious that {∥fF∥∣F∈ℱ(∧)}=∨{∥Tα∥∣α∈∧}, so we only need to prove that {∥fF∥∣F∈ℱ(∧)} is a.s. bounded, which is equivalent to the fact that {∥fF∥∣F∈ℱ(∧)} is 𝒯ϵ,λ-bounded in L+0(ℱ) by Proposition 16. Since ∨{fF(x)∣F∈ℱ(∧)}=∨{∥Tα(x)∥∣α∈∧} for each x∈S1, {fF(x)∣F∈ℱ(∧)} is a.s. bounded and hence also 𝒯ϵ,λ-bounded for each x∈S1. In the process of proof of sufficiency of (1), by replacing S2 with L0(ℱ,K) and the same reasoning we have that {∥fF∥∣F∈ℱ(∧)} is 𝒯ϵ,λ-bounded since {fF∣F∈ℱ(∧)} still satisfies all the three conditions of Proposition 28.Theorems30, 31, and 32 below are essentially known since they can be regarded as a special case of the classical closed graph theorem, open mapping theorem, and inverse operator theorem between Fréchét spaces only by noticing that a 𝒯ϵ,λ-complete RN space is a Fréchét space, but we would like to state them for the convenience of subsequent applications.Theorem 30.
Let(S1,∥·∥1) and (S2,∥·∥2) be 𝒯ϵ,λ-complete RN modules over K with base (Ω,ℱ,μ) and T:S1→S2 a module homomorphism. Then T is continuous from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ) if and only if T is 𝒯ϵ,λ-closed (namely, the graph of T is 𝒯ϵ,λ-closed in S1×S2).Theorem 31.
Let(S1,∥·∥1) and (S2,∥·∥2) be 𝒯ϵ,λ-complete RN modules over K with base (Ω,ℱ,μ) and T a surjective continuous module homomorphism from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ). Then T is 𝒯ϵ,λ-open; namely, T(G) is 𝒯ϵ,λ-open for each 𝒯ϵ,λ-open subset G of S1.Theorem 32.
Let(S1,∥·∥1) and (S2,∥·∥2) be 𝒯ϵ,λ-complete RN modules over K with base (Ω,ℱ,μ) and T a bijective continuous module homomorphism from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ). Then T-1 is also a continuous module homomorphism from (S2,𝒯ϵ,λ) to (S1,𝒯ϵ,λ).To give the versions of Theorems29 up to 32 under the locally L0-convex topology, let us first recall the notion of countable concatenation property of a set or an L0(ℱ,K)-module. The introducing of the notion utterly results from the study of the locally L0-convex topology, the reader will see that this notion is ubiquitous in the theory of the locally L0-convex topology; From now on, we always suppose that all the L0(ℱ,K)-modules E involved in this paper have the property that for any x,y∈E, if there is a countable partition {An,n∈N} of Ω to ℱ such that I~Anx=I~Any for each n∈N, then x=y. Guo already pointed out in [8] that all random locally convex modules possess this property, so the assumption is not too restrictive.Definition 33 (see [8]).
LetS be an L0(ℱ,K)-module. A subset G of S is said to have the countable concatenation property if for each sequence {gn,n∈N} in G and each countable partition {An,n∈N} of Ω to ℱ, there is g∈G such that I~Ang=I~Angn, for all n∈N.Two propositions below are key in this paper.Proposition 34 (see [8]).
Let(S,∥·∥) be an RN module and G a subset with the countable concatenation property. Then G-ϵ,λ=G-c, where G-ϵ,λ and G-c stand for the closures of G under the (ϵ,λ)-topology and the locally L0-convex topology, respectively.Proposition 35 (see [8]).
An RN module(S,∥·∥) is 𝒯ϵ,λ-complete if and only if it is 𝒯c-complete and S has the countable concatenation property.Theorem36 below has been used to establish random convex analysis; compare [12].Theorem 36.
Let(S1,∥·∥1) and (S2,∥·∥2) be two RN modules over K with base (Ω,ℱ,μ) such that S1 is 𝒯c-complete and has the countable concatenation property. Let {Tα:α∈∧} be a family of continuous module homomorphism from (S1,𝒯c) to (S2,𝒯c); then {Tα:α∈∧} is 𝒯c-bounded if and only if {Tα(x):α∈∧} is 𝒯c-bounded for each x∈S1.Proof.
By Proposition35, it follows from (2) of Theorem 29.Theorem 37.
Let(S1,∥·∥1) and (S2,∥·∥2) be two 𝒯c-complete RN modules over K with base (Ω,ℱ,μ) such that S1 and S2 have the countable concatenation property. Then, a module homomorphism T:S1→S2 is continuous from (S1,𝒯c) to (S2,𝒯c) if and only if  T is 𝒯c-closed (namely, the graph of  T is 𝒯c-closed in S1×S2).Proof.
It is clear that the graph ofT has the countable concatenation property. By Theorem 21, T is continuous from (S1,𝒯ϵ,λ) to (S2,𝒯ϵ,λ) if and only if it is continuous from (S1,𝒯c) to (S2,𝒯c). So, the proof follows from Propositions 34 and 35 and Theorem 30.Theorem 38.
Let(S1,∥·∥1) and (S2,∥·∥2) be two 𝒯c-complete RN modules over K with base (Ω,ℱ,μ) such that S1 and S2 have the countable concatenation property. If T:S1→S2 is a bijective continuous module homomorphism from (S1,𝒯c) to (S2,𝒯c), then T-1 is also continuous module homomorphism from (S2,𝒯c) to (S1,𝒯c).Proof.
It follows from Proposition35 and Theorems 21 and 32.To give Theorem40 below, we need Lemma 39 below.Lemma 39.
Let(S,∥·∥) be a 𝒯c-complete RN module over K with base (Ω,ℱ,μ) and M a 𝒯c-closed submodule of S such that both S and M have the countable concatenation property. Then, (S/M,∥·∥M) is still a 𝒯c-complete RN module and S/M has the countable concatenation property, where S/M is the quotient module of S with respect to M and ∥·∥M:S/M→L+0(ℱ) is defined by ∥x+M∥M=∧{∥y∥∣y∈x+M}.Proof.
By Proposition35 both S and M are 𝒯ϵ,λ-complete; then (S/M,∥·∥M) is a 𝒯ϵ,λ-complete RN module by the theory of quotient spaces for Fréchét spaces. The proof again follows from Proposition 35.Theorem 40.
Let(S1,∥·∥1) and (S2,∥·∥2) be two 𝒯c-complete RN modules over K with base (Ω,ℱ,μ) such that S1 and S2 have the countable concatenation property. If T is a surjective continuous module homomorphism from (S1,𝒯c) to (S2,𝒯c), then T is 𝒯c-open; namely, T(G) is 𝒯c-open for each 𝒯c-open subset G of S1.Proof.
LetM={x∈S1∣T(x)=0}; then M is 𝒯c-closed and has the countable concatenation property. Define T^:(S1/M,∥·∥M)→S2 by T^(x+M)=T(x), for all x∈S1, where (S1/M,∥·∥M) is the quotient space of (S1,∥·∥1) with respect to M; it is clear that T^ is a bijective continuous module homomorphism from (S1/M,𝒯c) to (S2,𝒯c). By Theorem 38, T^-1 is a continuous module homomorphism from (S2,𝒯c) to (S1/M,𝒯c). So, T^(G^) is a 𝒯c-open subset in S2 for each 𝒯c-open subset G^ of S1/M. Observing that T=T^∘J, where J:S1→S1/M is the canonical quotient mapping, then T is 𝒯c-open.Remark 41.
Since a𝒯c-complete RN module is not necessarily of second category, we can not obtain Theorem 40 by using the Baire category argument which is used in the proof of the classical open mapping theorem. In fact, the proof of Theorem 40 also gives a new proof of the classical open mapping theorem.


## 4. The Orthogonal Decomposition Theorem and Riesz Representation Theorem in Complete Random Inner Product Modules under the Two Kinds of Topologies

The orthogonal decomposition theorem in complete random inner product modules was already pointed out in [3, 20] without a detailed proof since it can be indirectly and similarly obtained from a best approximation result of [5, 21] in a special complete random inner product module. Here, we give it a detailed proof. The Riesz representation theorem in complete random inner product modules was proved in [20], but we did not strictly distinguish, by symbols, between measurable functions and their equivalence classes, so the readability of the proof given in [20] is not very good. Here, we also give a new proof for the sake of convenience for readers; the idea is, of course, due to [20].Definition 42 (see [3]).
An ordered pair(S,〈·,·〉) is called a random inner product space (briefly, an RIP space) over K with base (Ω,ℱ,μ) if S is a linear space over K and 〈·,·〉 is a mapping from S×S→L0(ℱ,K) such that the following are satisfied: (RIP-1)
〈x,x〉∈L+0(ℱ) and 〈x,x〉=0 implies x=0 (the null element of S);(RIP-2)
〈α·x,y〉=α·〈x,y〉, for all α∈K and x,y∈S;(RIP-3)
〈x,y〉=〈y,x〉¯, for all x,y∈S, where 〈y,x〉¯ stands for the complex conjugation of 〈y,x〉;(RIP-4)
〈x+y,z〉=〈x,z〉+〈y,z〉, for all x,y,z∈S, where 〈x,y〉 is called the random inner product of x and y in S.Furthermore, if, in addition,S is an L0(ℱ,K)-module and the following is satisfied: (RIPM-1)
〈ξ·x,y〉=ξ·〈x,y〉, for all ξ∈L0(ℱ,K) and x,y∈S, then (S,〈·,·〉) is called a random inner product module (briefly, an RIP module) over K with base (Ω,ℱ,μ), at which time 〈x,y〉 is called the L0-inner product of x and y in S; namely, an L0-inner product is a random inner product with the property (RIPM-1).In an RIP space(S,〈·,·〉), x is orthogonal to y, denoted by x⊥y, if 〈x,y〉=0. For a subset M of S, M⊥:={y∈S∣〈x,y〉=0,∀x∈M} is the orthogonal complement of M in S. Define ∥·∥:S→L+0(ℱ) by ∥x∥=〈x,x〉, for all x∈S; then (S,∥·∥) is an RN space over K with base (Ω,ℱ,μ) by the following random Schwartz inequality (namely, Lemma 43 below); ∥·∥ is the random norm derived from 〈·,·〉. It is also clear that (S,∥·∥) is an RN module if (S,〈·,·〉) is an RIP module.Lemma 43.
Let(S,〈·,·〉) be an RIP space over K with base (Ω,ℱ,μ). Then |〈x,y〉|≤∥x∥·∥y∥, for all x,y∈S.Proof.
Letx and y be fixed and then choose 〈x,y〉0, ∥x∥0, and ∥y∥0 as given representatives of 〈x,y〉, ∥x∥, and ∥y∥, respectively. Since 〈αx+y,αx+y〉=|α|2∥x∥2+2Re(α·〈x,y〉)+∥y∥2≥0, for all α∈K. Let K(1)={β∈K∣|β|=1}, then taking α=tβ with t∈R and β∈K(1) yields that t2·∥x∥2+2t·Re(β·〈x,y〉)+∥y∥2≥0, forallt∈R and β∈K(1); namely, t2·(∥x∥0(ω))2+2t·Re(β·〈x,y〉0(ω))+(∥y∥0(ω))2≥0a.s., forallt∈R and β∈K(1). Since R and K(1) are separable, we can obtain an ℱ-measurable Ω0 with μ(Ω∖Ω0)=0 such that t2·(∥x∥0(ω))2+2t·Re(β·〈x,y〉0(ω))+(∥y∥0(ω))2≥0 on Ω0, for all t∈R and β∈K(1).
For eachω∈Ω0, we can always take β∈K(1) such that β·〈x,y〉0(ω)=|〈x,y〉0(ω)|; then we have that t2·(∥x∥0(ω))2+2t·|〈x,y〉0(ω)|+(∥y∥0(ω))2≥0 on Ω0, for all t∈R, so |〈x,y〉0(ω)|≤∥x∥0(ω)·∥y∥0(ω), for all ω∈Ω0; namely, |〈x,y〉|≤∥x∥·∥y∥.Remark 44.
In the proof of Lemma43, we use a technique, namely, making use of separability of the scalar field K, which was first used in the proof of extension theorem for complex random linear functionals; compare [2, 8].Lemma 45.
Let(S,〈·,·〉) be an RIP space over K with base (Ω,ℱ,μ), M a subspace of M, and x0∈M. Then ∥x-x0∥=∧{∥x-y∥:y∈M} if and only if x-x0⊥M.Proof.
Sufficiency is obvious. As for the necessity, since∥x-x0∥2≤∥x-x0-αy∥2, for all α∈K and y∈M, namely, 2Re(α〈y,x-x0〉)≤|a|2∥y∥2, taking α=(1/n)a yields that 2Re(a〈y,x-x0〉)≤(1/n)|a|2∥y∥2, which implies that Re(a〈y,x-x0〉)≤0, for all a∈K and y∈M. Similar to the proof of Lemma 43, one can have that 〈y,x-x0〉=0, for all y∈M.Remark 46.
x
0 in Lemma 45 is called a best approximation point of x in M; such a kind of idea was earlier used in [5, 21–23] for the study of best approximation problems in Lebesgue-Bochner function spaces.Theorem 47.
Let(S,〈·,·〉) be a 𝒯ϵ,λ-complete RIP module over K with base (Ω,ℱ,μ) and M a 𝒯ϵ,λ-closed subspace of S. Then S=M⊕M⊥ if and only if M is a submodule.ProofSufficiency. For each x∈S, let d(x,M)=∧{∥x-y∥∣y∈M}. By Proposition 17 there exists a sequence {yn,n∈N} in M such that {∥x-yn∥,n∈N} converges a.s. to d(x,M) in a nonincreasing manner. Similar to the classical case, one can deduce that {yn,x∈N} is a 𝒯ϵ,λ-Cauchy sequence and hence convergent to some x0∈M such that ∥x-x0∥=d(x,M). By Lemma 45, x-x0⊥M. Hence, each x∈S can be written as x-x0+x0∈M⊥⊕M.Necessity. We only need to prove that I~Ax∈M for each A∈ℱ and x∈M. Let I~Ax=x1+x2 with x1∈M and x2∈M⊥; since x2⊥I~Ax implies x2=θ, I~Ax=x1∈M.Theorem 48.
Let(S,〈·,·〉) be a 𝒯ϵ,λ-complete RIP module over K with base (Ω,ℱ,μ). Then for each f∈S* there exists a unique yf∈S such that f(x)=〈x,yf〉, for all x∈S, and ∥f∥=∥yf∥.Before the proof of Theorem48, let us first introduce some notation and terminology as follows. Let ξ be in L0(ℱ,K) with a chosen representative ξ0. (ξ0)-1:Ω→K is defined by (ξ0)-1(ω)=1/ξ0(ω) if ξ0(ω)≠0 and 0 otherwise. Then the equivalence class determined by (ξ0)-1 is called the generalized inverse of ξ, denoted by ξ-1, and |ξ|-1ξ is called the sign of ξ, denoted by sgn(ξ). It is obvious that ξ·ξ-1=I~A, where A={ω∈Ω∣ξ0(ω)≠0}, and sgn(ξ)¯·ξ=|ξ|. Besides, for any ξ and η in L0(ℱ), [ξ≤η] denotes the equivalence class of the ℱ-measurable set {ω∈Ω∣ξ0(ω)≤η0(ω)}, where ξ0 and η0 are any chosen representatives of ξ and η, respectively. Similarly, one can understand the meaning of [ξ>η].We can now prove Theorem48.Proof of Theorem48.
Let Ran(f)={f(x)∣x∈S} and N(f)={x∈S∣f(x)=0}; then Ran(f) is a submodule of L0(ℱ,K) and N(f) a 𝒯ϵ,λ-closed submodule of S.
By Proposition17  {|f(x)|∣x∈S} is upward directed and hence there exists a sequence {xn,n∈N} in S such that {|f(xn)|,n∈N} converges a.s. to ξ:=∨{|f(x)|∣x∈S} in a nondecreasing manner.
Denoteξn=|f(xn)|, for all n∈N; let ξ0 and ξn0 be any chosen representatives of ξ and ξn, respectively, B={ω∈Ω∣ξ0(ω)>0}, and Bn={ω∈Ω∣ξn0(ω)>0}, foralln≥1. We can, without loss of generality, assume that Bn⊂Bn+1, for all n≥1, and ⋃n=1∞‍Bn=B. Further, let B0=⌀ and An=Bn∖Bn-1, for all n≥1; then Ai∩Aj=⌀, where i≠j, and B=⋃n=1∞‍An. Since ξn0>0 on An, we have f(zn)=I~An, where zn=I~An(f(xn))-1·xn, for all n≥1.
Ifμ(B)=0, then taking yf=θ ends the proof. If μ(B)>0, then f≠0, in which case we can assume that μ(An)>0, for all n∈N. By Theorem 47 there exists a unique z-n∈N(f)⊥ such that zn-z-n∈N(f), and hence f(z-n)=f(zn)=I~An, for all n∈N. Since I~Anx-I~Anf(x)z-n∈N(f), 〈I~Anx-I~Anf(x)z-n,z-n〉=0, foralln∈N and x∈S.
SinceI~An=f(z-n)=|f(z-n)|≤∥f∥∥z-n∥, ∥z-n∥>0 on An and I~Anf(x)=〈x,zn*〉, forallx∈S, where zn*=I~An(∥z-n∥)-1z-n, for all n∈N. Let yn=∑i=1n‍zi*, foralln∈N. By noticing that IBf(x)=f(x), for all x∈S, and I~B=limn→∞I~Bn=limn→∞∑i=1n‍I~Ai,f(x)=limn→∞(I~Bnf(x))=limn→∞(∑i=1n‍I~Aif(x))=limn→∞〈x,yn〉 (where convergence means the a.s. convergence). We can, without loss of generality, assume that (Ω,ℱ,μ) is a probability space; then μ{ω∈Ω∣∥yn+m-yn∥(ω)>ϵ}≤∑i=n+1n+m‍μ(Ai), for any ϵ>0 and n,m∈N, which implies that {yn,n∈N} is 𝒯ϵ,λ-Cauchy and hence convergent to some yf∈S, so f(x)=〈x,yf〉, for all x∈S.
∥
f
∥
≤
∥
y
f
∥ is obvious. Now, let A=[∥yf∥>0] and D=[|f(yf)|≤∥f∥∥yf∥]; then μ(D)=1 and IA∥f∥≥IA∥yf∥, where IA=I~A0 with A0 being any representative of A. On the other hand, (1-IA)∥f∥≥0=(1-IA)∥yf∥, so ∥f∥=IA∥f∥+(1-IA)∥f∥≥IA∥yf∥+(1-IA)∥yf∥=∥yf∥. Finally, the uniqueness of yf is obvious.The version of Theorem48 under the locally L0-convex topology, namely, Corollary 50 below, was given in [8]. Corollary 49 below is the version of Theorem 47 under the locally L0-convex topology.Corollary 49.
Let(S,∥·∥) be a 𝒯c-complete RIP module over K with base (Ω,ℱ,μ) and M a 𝒯c-closed submodule of S such that both S and M have the countable concatenation property. Then S=M⊕M⊥.Proof.
It follows from Propositions34 and 35 and Theorem 47.Corollary 50.
Let(S,〈·,·〉) be a 𝒯c-complete RIP module over K with base (Ω,ℱ,μ) such that S has the countable concatenation property. Then for each f∈S* there exists a unique yf∈S such that f(x)=〈x,yf〉, for all x∈S, and ∥f∥=∥yf∥.Proof.
It follows from Proposition35 and Theorem 48.Remark 51.
Based on Theorem48, we can establish the spectral representation theorem for random self-adjoint operators on complete complex random inner product modules, which has been used to establish the Stone’s representation theorem for a group of random unitary operators in [24].


## 5.(Lp(S))′≅Lq(S*)

In this section, let(S,∥·∥) be a given RN module over K with base (Ω,ℱ,μ) and S* its random conjugate space. Further, let 1≤p<+∞ and 1<q≤+∞ be a pair of Hölder conjugate numbers.Letr be an extended nonnegative number with 1≤r≤+∞ and (Lr(ℱ),|·|r) the Banach space of equivalence classes of r-integrable (when r<+∞) or essentially bounded (when r=+∞) real ℱ-measurable functions on (Ω,ℱ,μ) with the usual Lr-norm |·|r. Further, let Lr(S)={x∈S∣∥x∥∈Lr(ℱ)} and let ∥·∥r:Lr(S)→[0,+∞) be defined by ∥x∥r=|∥x∥|r, for all x∈Lr(S); then (Lr(S),∥·∥r) is a normed space over K and 𝒯ϵ,λ-dense in S; compare [22, 25]. Similarly, one can understand the implication of Lr(S*).Theorem52 below is proved in [25], a more general result is proved in [6] with Lr(ℱ) replaced by a Köthe function space, but the two proofs both only give the key idea of them. Here, we give a detailed proof of Theorem 52. Since our aim is to look for the tool for the development of the theory of RN modules together with their random conjugate spaces, Theorem 52 is enough for the aim.Theorem 52.
L
q
(
S
*
)
≅
(
L
p
(
S
)
)
′ under the canonical mapping T, where (Lp(S))′ denotes the classical conjugate space of Lp(S) and for each f∈Lq(S*), Tf (denoting T(f)): Lp(S)→K is defined by Tf(x)=∫Ω‍f(x)dμ, for all x∈Lp(S).We will divide the proof of Theorem52 into the following two Lemmas—Lemmas 53 and 54.Lemma 53.
T is isometric.Proof.
|
T
f
(
x
)
|
=
|
∫
Ω
‍
f
(
x
)
d
μ
|
≤
∫
Ω
|
f
(
x
)
|
‍
d
μ
≤
∫
Ω
‍
∥
f
∥
∥
x
∥
d
μ
≤
∥
f
∥
q
∥
x
∥
p, for all x∈Lp(S), so ∥Tf∥≤∥f∥q.
As for∥f∥q≤∥Tf∥, we can, without loss of generality, assume that (Ω,ℱ,μ) is a probability space. Let {xn,n∈N} be a sequence in S(1):={x∈S∣∥x∥≤1} such that {|f(xn)|,n∈N} converges a.s. to ∥f∥ in a nondecreasing manner (such a sequence {xn,n∈N} does exist!).
Whenp=1 and q=+∞, for any positive number ϵ let A(ϵ)=[∥f∥>∥f∥∞-ϵ]; then μ(A(ϵ))>0, and hence there exists some n0∈N such that B(ϵ):=[|f(xn0)|>∥f∥∞-ϵ] has a positive measure. Let xϵ=1/μ(B(ϵ))·IB(ϵ)·sgn(f(xn0))¯·xn0; then ∥xϵ∥1≤1 and |Tf(xϵ)|>∥f∥∞-ϵ, which shows that ∥Tf∥≥∥f∥∞.
Whenp>1,
(1)∫Ω‍|f(xn)|qdμ=∫Ω‍|f(xn)|q-1|f(xn)|dμ=∫Ω‍|f(xn)|q-1·sgn(f(xn))¯·f(xn)dμ=∫Ω‍f(|f(xn)|q-1·sgn(f(xn))¯·xn)dμ=Tf(|f(xn)|q-1·sgn(f(xn))¯·xn)≤∥Tf∥(∫Ω‍|f(xn)|qdμ)1/p,
then ∥f(xn)∥q≤∥Tf∥, for all n∈N. By the Levy theorem we have that ∥f∥q≤∥Tf∥.Lemma 54.
T is surjective.Proof.
For any fixedl∈(Lp(S))′ and x∈L∞(S), define the scalar measure Gx:ℱ→K and the vector measure G:ℱ→(L∞(S))′ as follows: Gx(E)=l(I~E·x), for all E∈ℱ,G(E)(y)=l(I~E·y), for all y∈L∞(S) and E∈ℱ.Since|G(E)(y)|=|l(I~E·y)|≤∥l∥·∥I~E·y∥p≤∥l∥·∥y∥∞·∥I~E∥p, for all E∈ℱ,y∈L∞(S), both G and Gx are countably additive. Now, for any finite partition {E1,E2,…,En} of Ω to ℱ and finitely many points x1,x2,…,xn, in the closed unit ball of L∞(S), we have |∑i=1n‍G(Ei)(xi)|=|l(∑i=1n‍I~Ei·xi)|≤∥l∥·|∑i=1n‍I~Ei|p=∥l∥. Similarly, we have that ∑i=1n‍|Gx(Ei)|=∑i=1n‍sgn(Gx(Ei))¯·Gx(Ei)=∑i=1n‍G(Ei)(sgn(Gx(Ei))¯·x)≤∥l∥·∥x∥∞. So |G|(Ω)≤∥l∥ and |Gx|(Ω)≤∥l∥·∥x∥∞; namely, G and Gx are both of bounded variation and they are both absolutely continuous with respect to μ.By the classical Radon-Nikodým theorem there exists a uniqueg(x)∈L1(ℱ,K) for each x∈L∞(S) such that Gx(E)=∫E‍g(x)dμ, for all E∈ℱ, and |Gx|(E)=∫E‍|g(x)|dμ, for all E∈ℱ, so we can obtain a mapping g:L∞(S)→L1(μ,K) such that (1)
g(αx+βy)=αg(x)+βg(y), for all α,β∈K, and x,y∈L∞(S);(2)
g(ξx)=ξ·g(x) for each simple element ξ in L0(ℱ,K),x∈L∞(S).We can now assert thatg(ξx)=ξg(x), for all ξ∈L∞(ℱ,K) and x∈L∞(S). In fact, for any ξ∈L∞(ℱ,K) there are always a sequence {ξn,n∈N} of simple elements in L0(ℱ,K) such that {∥ξn-ξ∥∞,n∈N} converges to 0; then ∥g(ξx)-g(ξnx)∥1=|G(ξ-ξn)x|(Ω)≤∥l∥·∥ξ-ξn∥∞·∥x∥∞→0(n→∞). On the other hand, ∥ξg(x)-ξng(x)∥1≤∥ξ-ξn∥∞∥g(x)∥1→0(n→∞), so g(ξx)=L1-limn→∞g(ξnx)=L1-limn→∞(ξng(x))=ξg(x).We prove that{|g(x)||x∈S(1)|} is upward directed as follows: for any x and y∈S(1), let E=[|g(x)|≤|g(y)|]; then |g(x)|∨|g(y)|=IE|g(y)|+(1-IE)|g(x)|=g(sgn(g(y))¯·IE·y+sgn(g(x))¯·(1-IE)·x)=g(z)=|g(z)|, where z=sgn(g(y))¯·IE·y+sgn(g(x))¯·(1-IE)·x∈S(1). Hence, there exists a sequence {xn,n∈N} in S(1) such that {|g(xn)|,n∈N} converges a.s. to ξ:=∨{|g(x)|∣x∈S(1)} in a nondecreasing manner. By the Levy theorem ∥ξ∥1=limn→∞∥g(xn)∥1=limn→∞|Gxn|(Ω)≤|G|(Ω)≤∥l∥<+∞, so ξ∈L1(ℱ,K). Since for any positive number ϵ and x∈L∞(S), it is clear that |g((1/(∥x∥+ϵ))x)|≤ξ; namely, |g(x)|≤ξ(∥x∥+ϵ), which implies that |g(x)|≤ξ∥x∥, for all x∈L∞(S). By the Hahn-Banach theorem for a.s. bounded random linear functionals (see [2, 8]) there is an a.s. bounded random linear functional f∈S* such that f∣L∞(S)=g; further f is unique since L∞(S) is 𝒯ϵ,λ-dense in S and ∥f∥≤ξ; we also have that ∥f∥∈L1(ℱ,K).By the definition ofGx, l(x)=Gx(Ω)=∫Ω‍g(x)dμ=∫Ω‍f(x), for all x∈L∞(S). We prove that f∈Lq(S*) as follows: let En=[∥f∥≤n] and fn=IEn·f; then fn∈Lq(S*) and ∥fn∥=IEn∥f∥ (here, we can assume that μ is a probability measure). Since ∫En‍f(x)dμ=l(IEn·x), for all x∈L∞(S) and n∈N, then Tfn∣L∞(S)=l∣IE·L∞(S). From the fact that L∞(S) is dense in (Lp(S),∥·∥p), Tfn∣Lp(S)=l∣IEn·Lp(S), so ∥Tfn∥≤∥l∥; letting n→+∞ yields that |∥f∥|q≤limn→∞∥fn∥q=limn→∞∥Tfn∥≤∥l∥<+∞; namely, f∈Lq(S*). Further, we also have that l=Tf since they coincide on the dense subspace L∞(S) of Lp(S).Remark 55.
Let(B,∥·∥) be a normed space over K and L0(ℱ,B) the L0(ℱ,K)-module of equivalence classes of B-valued ℱ-strongly measurable functions on (Ω,ℱ,μ). Let B′ be the classical conjugate space of B and L0(ℱ,B′,w*) the L0(ℱ,K)-module of w*-equivalence classes of B′-valued w*-measurable functions on (Ω,ℱ,μ). For any x∈L0(ℱ,B) with a representative x0, the L0-norm of x is defined to be the equivalence class of ∥x0∥, still denoted by ∥x∥; then (L0(ℱ,B),∥·∥) is an RN module over K with base (Ω,ℱ,μ). For any y∈L0(ℱ,B′,w*) with a representative y0, the L0-norm of y is defined to be the equivalence class of esssup{|y0(b)|∣b∈Band∥b∥≤1} (namely, the essential supremum of {|y0(b)|∣b∈Band∥b∥≤1}); then L0(ℱ,B′,w*) is also an RN module over K with base (Ω,ℱ,μ). In [26] it is proved that (L0(ℱ,B))*=L0(ℱ,B′,w*), so if we take S=L0(ℱ,B) in Theorem 52 then Lp(ℱ,B)′≅Lq(ℱ,B′,w*). Generally speaking, σ-finite measure spaces are enough for various kinds of problems in analysis, but some more general measure spaces are sometimes necessary; for example, strictly localizable measure spaces are considered in [6]. Even in [27] we introduced the notion of an RN module with base being an arbitrary measure space (Ω,ℱ,μ) by defining it to be a projective limit of a family of RN modules with base (A,A∩ℱ,μ∣A∩ℱ), where A∈ℱ satisfies 0<μ(A)<+∞, and further proved that Theorem 52 remains true for any measure space, so Theorem 52 unifies all the representation theorems of the dual spaces of Lebesgue-Bochner function spaces. As said in [6, 25], it is more interesting that Theorem 52 establishes the key connection between random conjugate spaces and classical conjugate spaces, which has played a crucial role in the subsequent development of random conjugate spaces; compare [22, 28, 29].Remark 56.
Since the Lebesgue-Bochner function spaceLp(ℱ,B) (or is written as Lp(μ,B)) has the target space B, the simple functions in Lp(ℱ,B) always play an active role in the study of the dual of Lp(ℱ,B), whereas we do not have the counterparts of simple elements in Lp(ℱ,B) for abstract spaces Lp(S), so we are forced to replace simple elements in Lp(ℱ,B) with elements in L∞(S) in order to complete the proof of Theorem 52. In [26] we prove that a Banach space B is reflexive if and only if L0(ℱ,B) is random reflexive; the original motivation of Theorem 52 is to establish the following characterization.Theorem 57.
Let(S,∥·∥) be a 𝒯ϵ,λ-complete RN module over K with base (Ω,ℱ,μ) and p any given positive number such that 1<p<+∞. Then S is random reflexive if and only if Lp(S) is reflexive.Proof.
LetJ:S→S** and j:Lp(S)→(Lp(S))′′ be the corresponding canonical embedding mappings.(1) Necessity. Since 1<p<+∞, its Hölder conjugate number q satisfies 1<q<+∞; then (Lp(S))′′=(Lq(S*))′=Lp(S**)=Lp(S).(2) Sufficiency. Let f** be any given element in S** and En=[n-1≤∥f**∥<n] for any n∈N. We can, without loss of generality, assume that μ is a probability measure; then ∑n=1∞‍μ(En)=1. Since IEnf**∈Lp(S**)=(Lp(S))′′, there exists xn∈Lp(S) such that j(xn)=IEnf**; namely, for each f*∈Lq(S*), we have that ∫Ω‍f*(xn)dμ=j(xn)(f*)=∫Ω‍IEnf**(f*)dμ. By replacing f* with IEf* we can obtain that ∫E‍f*(xn)dμ=∫E‍IEnf**(f*)dμ, for all E∈ℱ and f*∈Lq(S*), which implies that f*(xn)=IEnf**(f*), for all f*∈Lq(S*). Since Lq(S*) is 𝒯ϵ,λ-dense in S*, J(xn)=IEnf**, foralln∈N.Letyn=∑k=1n‍IEk·xk, for all n∈N; then {yn,n∈N} is 𝒯ϵ,λ-Cauchy in S and hence convergent to some y∈S, which shows that J(y)=limn→∞J(yn)=limn→∞∑k=1n‍IEk·f**=(∑n=1∞‍IEn)·f**=f**; namely, J is surjective.Remark 58.
Concerning Theorem57, a similar and more general result was given in [6] where Lp(ℱ) is replaced by a reflexive Köthe function space.



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