#### 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 -topology and the locally -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, , where and are a pair of Hölder conjugate numbers with a random normed module, the random conjugate space of the corresponding (resp., ) space derived from (resp., ), and the ordinary conjugate space of

#### 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 -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 -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 -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 -topology and the locally -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 -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 -convex topology.

The remainder of this paper is organized as follows. In Section 2, 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, .

#### 2. Preliminaries

Throughout this paper, denotes the scalar field, namely, the field of real numbers or the field of complex numbers, a -finite measure space, the algebra of equivalence classes of -measurable -valued functions on , the set of equivalence classes of extended real-valued -measurable functions on and .

is partially ordered by if and only if *a.s.*, where and are arbitrarily chosen representatives of and in , respectively. It is well known from [13] that every subset in has a supremum and infimum, denoted by and , respectively, and there are countable subsets and of such that and . Furthermore, if, in addition, is upward directed or downward directed, then and can be chosen as nondecreasing and nonincreasing, respectively. In particular, is conditionally complete, namely, every subset with an upper bound has a supremum.

Following are the notation and terminology frequently used in this paper: ,
,
,
where “ on ” means that *a.s.* for an arbitrarily chosen representative of .

*Definition 1 (see [3]). * An ordered pair is called a random normed space (briefly, an RN space) over with base if is a linear space over and is a mapping from to such that the following three conditions are satisfied: (RN-1) implies (the null in ), (RN-2) , for all and , (RN-3) , for all ,where is called the random norm of . If only satisfies (RN-2) and (RN-3), then it is called a random seminorm.

Furthermore, if, in addition, is a left module over the algebra (briefly, an -module) and the following additional condition is also satisfied:(RNM-1) , for all and , then is called a random normed module (briefly, an RN module) over with base , at which time is called an norm on . Similarly, if only satisfies (RN-3) and (RNM-1), then it is called an -seminorm on .

*Remark 2. *In [1], the original definition of an RN space was given by only requiring to be a probability space and defining to be a nonnegative random variable; the corresponding (RN-1) to (RN-3) are given in the following way:(RN-1)′ *a.s.* implies , (RN-2)′ *a.s.*, for all and ,(RN-3)′ *a.s.*, for all .

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 Definition 1 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 is termed as a randomly normed -module in [6] and an -normed module in [7]. We still would like to continue to employ the terminology “an RN module over with base ” in order to keep concordance with the earliest terminology used in [1].

*Definition 4 (see [3, 5, 14]). * Let and be two RN spaces over with base . A linear operator from to is said to be *a.s.* bounded if there exists such that , in which case is defined to be , called the random norm of . Denote the linear space of *a.s.* bounded linear operators from to by ; then still becomes an RN space over with base when is defined as above for every . In particular, when and (namely, the absolute value mapping), is called the random conjugate space of and an element in is called an *a.s.* bounded random linear functional on .

*Remark 5. *When in Definition 4 is an RN module, automatically becomes an RN module under the module operation , for all , and . When and are both RN modules, in [6] is used to stand for the -module of *a.s.* bounded module homomorphisms from to ; we will show that in the special case an *a.s.* bounded linear operator must be a module homomorphism. Therefore, the two implications of coincide in this case.

As in the classical functional analysis, we can similarly define a conjugate operator for an *a.s.* bounded linear operator from to as follows: , for all . From the Hahn-Banach theorem for *a.s.* bounded random linear functional established in [2] (also see [8]), one has that .

For the sake of convenience, let us recall some notation and terminology in the theory of probabilistic normed spaces.

*Definition 6 (see [1]). * A function is called a weak -norm if the following are satisfied: (t-1) , for all ,(t-2) , for all with ,(t-3) .A weak -norm is called a -norm if the following two additional conditions are satisfied:(t-4) , for all ,(t-5) , for all .

Although -norms are widely used in the theory of probabilistic metric spaces, weak -norms have their own advantages, for example, for a family of weak -norms, defined by , for all , is still a weak -norm, whereas this is not true for -norms.

Throughout this paper, is nondecreasing and left continuous on , and , and . For extended real random variable on a probability space , its (left continuous) distribution function is defined by , for all .

In particular, stands for the distribution function defined by when and when ; namely, is the distribution function of the constant .

*Definition 7 (see [1]). * A triple is called a Menger probabilistic normed space (briefly, an M-PN space) over if is a linear space over is a mapping from to , and is a weak -norm such that the following are satisfied:(MPN-1) if and only if (the null element in ),(MPN-2) , for all and ,(MPN-3) , for all and . Here is called the probabilistic norm of .

For an M-PN space , let is a weak -norm such that , and define by , for all ; then it is very easy to see that . is called the largest weak -norm of such that is an M-PN space under . From now on, for an M-PN space , we always assume that is the largest weak -norm of .

In [15], LaSalle introduced the notion of a pseudonormed linear space: let be a linear space over and a family of mappings from to and indexed by a directed set ; then is called a pseudonormed linear space if the following are satisfied:(PNS-1) , for all , and ,(PNS-2) , for all such that ,(PNS-3) for any , there exists such that , for all .

Let be a pseudonormed linear space. For any and , let . Then is a local base at the null element of some linear topology for , called the linear topology induced by . Conversely any linear topology for can be induced by some such that is a pseudonormed linear space.

To connect an M-PN space to a pseudonormed linear space, for each , define by , for all . Then we have the following.

Theorem 8 (see [16]). * Let be an M-PN space. Then one has the following statements.*(1)* if and only if is a pseudonormed linear space; namely, for each there exists such that , for all .*(2)*, namely, , for all , if and only if is a seminorm on for each ; namely, is a -type space.*(3)* for all such that if and only if there exists a norm on such that , for all .*

Theorem 8 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 be an M-PN space such that . For any positive numbers and with , let ; then forms a local base at of some metrizable linear topology for , called the -topology induced by .*

From Theorem 8, one can easily see that the -topology for an M-PN space with is exactly the one induced by the family 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 with and the theory of pseudonormed linear space 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 . 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 be an M-PN space with and a subset of is defined by , for all , and , called the probabilistic diameter of . If , then is said to be probabilistically bounded.

Proposition 11 below is a straightforward verification by definition.

Proposition 11. *Let and be the same as in Definition 10. Then is probabilistically bounded if and only if is bounded with respect to the -topology (namely, can be absorbed by every -neighborhood of the null ).*

Let be a probability space and an RN space over with base . Define by and ; namely, is the distribution of ; then is an M-PN space with , where , for all is called the M-PN space determined by ; the -topology for is also called the -topology for .

When is a -finite measure space, let ; then the following definition is a slight generalization of the case when is a probability space.

*Definition 12 (see [3]). *Let be an RN space over with base . For and , let . Then forms a local base at of some metrizable linear topology for , called the -topology for induced by .

Proposition 13 below is a straightforward verification by definition.

Proposition 13. *Let be an RN space over with base and a countable partition of to such that . Then one has the following.*(1)*A sequence ** in ** converges in the **-topology to ** in ** if and only if ** converges to *
0
* locally in measure; namely, ** converges to *
0
* in measure ** on each *. (2)*The **-topology for ** is exactly the linear topology induced by the quasinorm ** defined by ** for all *. (3)*Let ** be defined by **; then ** is a probability measure equivalent to ** and ** has the same **-topology whether ** is regarded as an RN space with base ** or *.

*Remark 14. *When is a -finite measure space, the -topology for the special RN space 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 -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 and are studied. Finally, independently of B Schweizer and Sklar’s work [1], the -topology is also introduced in [6], called the -topology.

*Definition 15 (see [3, 5, 14]). * Let be an RN space over with base and a subset of is said to be *a.s.* bounded if .

In the sequel, the -topology for every RN space is always denoted by 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 be an RN space with base and a subset of such that is upward directed. Then is a.s. bounded if and only if it is bounded, at which time and when is a probability space, , where and is the distribution function of .*

*Proof. *We can, without loss of generality, assume that is a probability space. Necessity is clear. We prove the sufficiency as follows.

Since there exists a sequence in such that converges *a.s.* to in a nondecreasing manner. Let be the M-PN space determined by ; then converges weakly to ; it is easy to check that (namely, the probabilistic diameter of ), and hence . But is clear, then . Since is -bounded, , which shows that .

Proposition 17 below gives a very general condition for to be upward directed or downward directed.

Proposition 17. *Let be an RN module with base and a subset of such that for any , where and stands for the equivalence class of the characteristic function of . Then is both upward and downward directed. *

* Proof. *We only prove that is upward directed; the case of being downward directed is similar. For any , let , where and are arbitrarily chosen representatives of and , respectively. Then is such that . Since , the proof is complete.

It is easy to see that is a topological algebra over and is a topological module over when is an RN module over with base . In 2009, another kind of topology for an RN module was introduced in [7].

*Definition 18 (see [7]). * Let be an RN module over with base . A subset of is called a -open set if for each there exists some such that , where . Denote by the family of -open sets; then forms a Hausdorff topology for , called the locally -convex topology induced by .

It is easy to check that the locally -convex topology is much stronger than the -topology for a given RN module; is, however, only a topological ring since it is unnecessarily a linear topological space (see [7]). Furthermore, for an RN module over with base is a topological module over the topological ring compare [7]. From now on, the locally -convex topology for every RN module is always denoted by when no confusion occurs.

*Definition 19. *Let be an -module. A subset of is said to be -convex if , for all and with . A subset of is said to be -balanced if for all and with . A subset of is said to be -absorbed by a subset of if there exists some such that for all with . Furthermore, if a subset of -absorbs every element of , then is said to be -absorbent.

*Definition 20 (see [12]). * Let be an RN module and a subset of is said to be -bounded if is -absorbed by every -neighborhood of the null element.

It is also very easy to see that a subset of an RN module is -bounded if and only if it is *a.s.* bounded.

For the sake of convenience, always denotes the characteristic function of and the equivalence class of . As usual, is called the equivalence class of , denoted by ; we sometimes also use for .

Theorem 21 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 the M-PN space determined by a given RN space .

Theorem 21. *Let and be two RN modules over with base and a linear operator. Then one has the following: *(1)* is a.s. bounded if and only if is a continuous module homomorphism from to ; *(2)

*is*

*a.s.*bounded if and only if is a continuous module homomorphism from to .*Proof*

*(1) Necessity*. Since is *a.s.* bounded, must be continuous from to ; it remains to prove that is also a module homomorphism; it suffices to prove that , for all and , since is linear. Since , , for all . Then , for all . On the other hand, , for all and for all . So, .

*Sufficiency*. is *a.s.* bounded, and hence also -bounded; further is -bounded since is a continuous linear operator. Besides, for all since has this property and is a module homomorphism. Then is *a.s.* bounded; namely, by Propositions 16 and 17. Since , for all and , , which implies that , for all ; that is to say, is *a.s.* bounded, at which time it is also clear that .

*(2) Necessity*. From the proof of necessity of (1), if is *a.s.* bounded then is a module homomorphism. The fact that is *a.s.* bounded also obviously implies that is continuous from to .

*Sufficiency*. Since is a -neighborhood of the null element of there exists some such that , where . Thus for any , , for all , and ; namely, by the fact that is a module homomorphism, which shows that , for all ; namely, is *a.s.* bounded.

*Remark 22. * 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, ], one careful reader can see that 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 and be two RN modules over with base ; when is a continuous module homomorphism from to or from to , the process of proof of Theorem 21 has implied that ; further we have by Proposition 16, where is the probabilistic norm of , namely, the distribution function of , and is the probabilistic diameter of .

The proof of Proposition 24 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 be an RN module over with base and such that the following two conditions are satisfied: *(1)*, for all ** and all nonnegative numbers **;*(2),
* for all **. **Then is a.s. bounded; namely, there is some such that , for all, if and only if is continuous from to and and , at which time , where ; furthermore if, in addition, is a probability space, then (the distribution function of . *

It is well known that is a Banach space when and are normed spaces and is complete. Similarly, is -complete when and are RN modules and is -complete, which is independently pointed out by Guo in [5, 14] and Haydon et al. in [6]; in particular is -complete for every RN module . In fact, a more general result is proved in [14], namely, the following.

Proposition 25 (see [14]). * Let and be two RN spaces over with base such that is -complete; then is -complete. *

When and are both RN modules, since , for any , the proof of Proposition 25 is similar to that of the classical case. But when and are only RN spaces, its proof needs Lemma 26 below. To state it, let us recall the canonical embedding mapping from an RN space to , where is defined by , for all and . It is easy to see that is random-norm preserving. As usual, is said to be random reflexive if is surjective. Generally, the -closed submodule generated by in is called the -closed submodule generated by , denoted by ; it is, obviously, a -complete RN module.

Lemma 26 below is given and proved in [14]; here we give it a better proof.

Lemma 26 (see [14]). *Let and be two RN spaces over with base such that is -complete. Then is isomorphic to a -closed subspace of in a random-norm-preserving way.*

* Proof. *Define by , where is the random conjugate operator of .

First, is well defined, namely, , and isometric. Let and be the corresponding canonical embedding mappings; it is easy to check that , which not only shows that but also shows that . Since . Further by (1) of Theorem 21 one can easily see that .

Second, is a -closed subspace of . Let be a sequence in such that converges in the -topology to some . Then is also -Cauchy in . We can, without loss of generality, assume that converges *a.s.* to some . Define by ; then is well defined since is -complete, and is *a.s.* bounded since , for all . Finally, it is easy to check that .

*Remark 27. *Since is always -complete, so is when is -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 -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 -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 -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]). * Let be a linear topological space over of second category and a quasinormed linear space. Let be a family of continuous mappings from to such that the following three properties are satisfied:*(1),
* for all ** and **;*(2),
* for all **, and **;*(3)* is bounded with respect to the linear topology induced by ** for each **.** Then uniformly in . *

Theorem 29. *Let and be two RN modules over with base such that is -complete. Let . be a family of continuous module homomorphisms from to . Then, one has the following: *(1)* is -bounded in if and only if is -bounded in for each ; *(2)* is a.s. bounded in if and only if is a.s. bounded in for each . *

* Proof. *We can, without loss of generality, assume that is a probability space.

*(1) Necessity*. Let be -bounded in ; namely, . For each , since , ; namely, , for all . Then, , for all and , where , for all . Since for any , then ; namely, is -bounded in for each .

*Sufficiency*. Let be defined by for any ; then is a quasinormed linear space and induces the -topology for . Since is a linear topological space of the second category and it is also clear that satisfies all the three conditions of Proposition 28, uniformly in by Proposition 28, which certainly implies that is -bounded in for each -bounded set in , in particular is -bounded. By Remark 23, , for all . Define by , for all , and by , and , ; denote by ; then one can easily check that ; then is -bounded since .

*(2) Necessity of (2) Is Clear*. We prove sufficiency of (2) as follows.

Denote the family of finite subsets of by . For any , define by , for all ; then is continuous from to and , for all and , and hence is *a.s.* bounded and . It is obvious that , so we only need to prove that is *a.s.* bounded, which is equivalent to the fact that is -bounded in by Proposition 16. Since for each , is *a.s.* bounded and hence also -bounded for each . In the process of proof of sufficiency of (1), by replacing with and the same reasoning we have that is -bounded since still satisfies all the three conditions of Proposition 28.

Theorems 30, 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 -complete RN space is a Fréchét space, but we would like to state them for the convenience of subsequent applications.

Theorem 30. *Let and be -complete RN modules over with base and a module homomorphism. Then is continuous from to if and only if is -closed (namely, the graph of is -closed in ). *

Theorem 31. *Let and be -complete RN modules over with base and a surjective continuous module homomorphism from to . Then is -open; namely, is -open for each -open subset of . *

Theorem 32. *Let and be -complete RN modules over with base and a bijective continuous module homomorphism from to . Then is also a continuous module homomorphism from to .*

To give the versions of Theorems 29 up to 32 under the locally -convex topology, let us first recall the notion of countable concatenation property of a set or an -module. The introducing of the notion utterly results from the study of the locally -convex topology, the reader will see that this notion is ubiquitous in the theory of the locally -convex topology; From now on, we always suppose that all the -modules involved in this paper have the property that for any , if there is a countable partition of to such that for each , then . 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]). * Let be an -module. A subset of is said to have the countable concatenation property if for each sequence in and each countable partition of to , there is such that , for all .

Two propositions below are key in this paper.

Proposition 34 (see [8]). *Let be an RN module and a subset with the countable concatenation property. Then , where and stand for the closures of under the -topology and the locally -convex topology, respectively. *

Proposition 35 (see [8]). * An RN module is -complete if and only if it is -complete and has the countable concatenation property. *

Theorem 36 below has been used to establish random convex analysis; compare [12].

Theorem 36. *Let and be two RN modules over with base such that is -complete and has the countable concatenation property. Let be a family of continuous module homomorphism from to ; then is -bounded if and only if is -bounded for each . *

*Proof. *By Proposition 35, it follows from (2) of Theorem 29.

Theorem 37. *Let and be two -complete RN modules over with base such that and have the countable concatenation property. Then, a module homomorphism is continuous from to if and only if is -closed (namely, the graph of is -closed in ). *

*Proof. * It is clear that the graph of has the countable concatenation property. By Theorem 21, is continuous from to if and only if it is continuous from to . So, the proof follows from Propositions 34 and 35 and Theorem 30.

Theorem 38. *Let and be two -complete RN modules over with base such that and have the countable concatenation property. If is a bijective continuous module homomorphism from to , then is also continuous module homomorphism from to . *

* Proof. *It follows from Proposition 35 and Theorems 21 and 32.

To give Theorem 40 below, we need Lemma 39 below.

Lemma 39. *Let be a -complete RN module over with base and a -closed submodule of such that both and have the countable concatenation property. Then, is still a -complete RN module and has the countable concatenation property, where is the quotient module of with respect to and is defined by . *

*Proof. *By Proposition 35 both and are -complete; then is a -complete RN module by the theory of quotient spaces for Fréchét spaces. The proof again follows from Proposition 35.

Theorem 40. * Let and be two -complete RN modules over with base such that *