Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 989102, 13 pages

http://dx.doi.org/10.1155/2013/989102

Research Article

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

Guo Tiexin

School of Mathematics and Statistics, Central South University, Hunan Province, Changsha 410075, China

Received 24 April 2013; Accepted 22 July 2013

Academic Editor: Pei De Liu

Copyright © 2013 Guo Tiexin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 and have the countable concatenation property. If is a surjective continuous module homomorphism from to , then is -open; namely, is -open for each -open subset of .

Proof. Let ; then is -closed and has the countable concatenation property. Define by , for all , where is the quotient space of with respect to ; it is clear that is a bijective continuous module homomorphism from to . By Theorem 38, is a continuous module homomorphism from to . So, is a -open subset in for each -open subset of . Observing that , where is the canonical quotient mapping, then is -open.

Remark 41. Since a -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 is called a random inner product space (briefly, an RIP space) over with base if is a linear space over and is a mapping from such that the following are satisfied:

 (RIP-1)  and implies (the null element of ); (RIP-2) , for all and ; (RIP-3) , for all , where stands for the complex conjugation of ;  (RIP-4) , for all ,

where is called the random inner product of and in .

Furthermore, if, in addition, is an -module and the following is satisfied:

(RIPM-1) , for all and ,

then is called a random inner product module (briefly, an RIP module) over with base , at which time is called the -inner product of and in ; namely, an -inner product is a random inner product with the property (RIPM-1).

In an RIP space , is orthogonal to , denoted by , if . For a subset of , is the orthogonal complement of in . Define by , for all ; then is an RN space over with base by the following random Schwartz inequality (namely, Lemma 43 below); is the random norm derived from . It is also clear that is an RN module if is an RIP module.

Lemma 43. Let be an RIP space over with base . Then , for all .

Proof. Let and be fixed and then choose , , and as given representatives of , , and , respectively. Since , for all . Let , then taking with and yields that , and ; namely, a.s., and . Since and are separable, we can obtain an -measurable with such that on , for all and .

For each , we can always take such that ; then we have that on , for all , so , for all ; namely, .

Remark 44. In the proof of Lemma 43, we use a technique, namely, making use of separability of the scalar field , which was first used in the proof of extension theorem for complex random linear functionals; compare [2, 8].

Lemma 45. Let be an RIP space over with base , a subspace of , and . Then if and only if .

Proof. Sufficiency is obvious. As for the necessity, since , for all and , namely, , taking yields that , which implies that , for all and . Similar to the proof of Lemma 43, one can have that , for all .

Remark 46. in Lemma 45 is called a best approximation point of in ; 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 be a -complete RIP module over with base and a -closed subspace of . Then if and only if is a submodule.

Proof

Sufficiency. For each , let . By Proposition 17 there exists a sequence in such that converges a.s. to in a nonincreasing manner. Similar to the classical case, one can deduce that is a -Cauchy sequence and hence convergent to some such that . By Lemma 45, . Hence, each can be written as .

Necessity. We only need to prove that for each and . Let with and ; since implies , .

Theorem 48. Let be a -complete RIP module over with base . Then for each there exists a unique such that , for all , and .

Before the proof of Theorem 48, let us first introduce some notation and terminology as follows. Let be in with a chosen representative . is defined by if and otherwise. Then the equivalence class determined by is called the generalized inverse of , denoted by , and is called the sign of , denoted by sgn. It is obvious that , where , and . Besides, for any and in , denotes the equivalence class of the -measurable set , where and are any chosen representatives of and , respectively. Similarly, one can understand the meaning of .

We can now prove Theorem 48.

Proof of Theorem 48. Let Ran and ; then Ran is a submodule of and a -closed submodule of .

By Proposition 17   is upward directed and hence there exists a sequence in such that converges a.s. to in a nondecreasing manner.

Denote , for all ; let and be any chosen representatives of and , respectively, , and , . We can, without loss of generality, assume that , for all , and . Further, let and , for all ; then , where , and . Since on , we have , where , for all .

If , then taking ends the proof. If , then , in which case we can assume that , for all . By Theorem 47 there exists a unique such that , and hence , for all . Since , , and .

Since , on and , , where , for all . Let , . By noticing that , for all , and (where convergence means the a.s. convergence). We can, without loss of generality, assume that is a probability space; then , for any and , which implies that is -Cauchy and hence convergent to some , so , for all .

is obvious. Now, let and ; then and , where with being any representative of . On the other hand, , so . Finally, the uniqueness of is obvious.

The version of Theorem 48 under the locally -convex topology, namely, Corollary 50 below, was given in [8]. Corollary 49 below is the version of Theorem 47 under the locally -convex topology.

Corollary 49. Let be a -complete RIP module over with base and a -closed submodule of such that both and have the countable concatenation property. Then .

Proof. It follows from Propositions 34 and 35 and Theorem 47.

Corollary 50. Let be a -complete RIP module over with base such that has the countable concatenation property. Then for each there exists a unique such that , for all , and .

Proof. It follows from Proposition 35 and Theorem 48.

Remark 51. Based on Theorem 48, 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.

In this section, let be a given RN module over with base and its random conjugate space. Further, let and be a pair of Hölder conjugate numbers.

Let be an extended nonnegative number with and the Banach space of equivalence classes of -integrable (when ) or essentially bounded (when ) real -measurable functions on with the usual -norm . Further, let and let be defined by , for all ; then is a normed space over and -dense in ; compare [22, 25]. Similarly, one can understand the implication of .

Theorem 52 below is proved in [25], a more general result is proved in [6] with 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. under the canonical mapping , where denotes the classical conjugate space of and for each , (denoting ): is defined by , for all .

We will divide the proof of Theorem 52 into the following two Lemmas—Lemmas 53 and 54.

Lemma 53. is isometric.

Proof. , for all , so .

As for , we can, without loss of generality, assume that is a probability space. Let be a sequence in such that converges a.s. to in a nondecreasing manner (such a sequence does exist!).

When and , for any positive number let ; then , and hence there exists some such that has a positive measure. Let ; then and , which shows that .

When ,

then , for all . By the Levy theorem we have that .

Lemma 54. is surjective.

Proof. For any fixed and , define the scalar measure and the vector measure as follows:

 , for all , , for all and .

Since , for all , both and are countably additive. Now, for any finite partition of to and finitely many points , in the closed unit ball of , we have . Similarly, we have that . So and ; namely, and are both of bounded variation and they are both absolutely continuous with respect to .

By the classical Radon-Nikodým theorem there exists a unique for each such that , for all , and , for all , so we can obtain a mapping such that

(1), for all , and ; (2) for each simple element in .

We can now assert that , for all and . In fact, for any there are always a sequence of simple elements in such that converges to ; then . On the other hand, , so .

We prove that is upward directed as follows: for any and , let ; then , where . Hence, there exists a sequence in such that converges a.s. to in a nondecreasing manner. By the Levy theorem , so . Since for any positive number and , it is clear that ; namely, , which implies that , for all . By the Hahn-Banach theorem for a.s. bounded random linear functionals (see [2, 8]) there is an a.s. bounded random linear functional such that ; further is unique since is -dense in and ; we also have that .

By the definition of , , for all . We prove that as follows: let and ; then and (here, we can assume that is a probability measure). Since , for all and , then . From the fact that is dense in , , so ; letting yields that ; namely, . Further, we also have that since they coincide on the dense subspace of .

Remark 55. Let be a normed space over and the -module of equivalence classes of -valued -strongly measurable functions on . Let be the classical conjugate space of and the -module of -equivalence classes of -valued -measurable functions on . For any with a representative , the -norm of is defined to be the equivalence class of , still denoted by ; then is an RN module over with base . For any with a representative , the -norm of is defined to be the equivalence class of (namely, the essential supremum of ); then is also an RN module over with base . In [26] it is proved that , so if we take in Theorem 52 then . Generally speaking, -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 by defining it to be a projective limit of a family of RN modules with base , where satisfies , 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 space (or is written as ) has the target space , the simple functions in always play an active role in the study of the dual of , whereas we do not have the counterparts of simple elements in for abstract spaces , so we are forced to replace simple elements in with elements in in order to complete the proof of Theorem 52. In [26] we prove that a Banach space is reflexive if and only if is random reflexive; the original motivation of Theorem 52 is to establish the following characterization.

Theorem 57. Let be a -complete RN module over with base and any given positive number such that . Then is random reflexive if and only if is reflexive.

Proof. Let and be the corresponding canonical embedding mappings.

(1) Necessity. Since , its Hölder conjugate number satisfies ; then .

(2) Sufficiency. Let be any given element in and for any . We can, without loss of generality, assume that is a probability measure; then . Since , there exists such that ; namely, for each , we have that . By replacing with we can obtain that , for all and , which implies that , for all . Since is -dense in , , .

Let , for all ; then is -Cauchy in and hence convergent to some , which shows that ; namely, is surjective.

Remark 58. Concerning Theorem 57, a similar and more general result was given in [6] where is replaced by a reflexive Köthe function space.

Acknowledgment

The author would like to thank Professor Quanhua Xu for kindly providing the reference [6] in February 2012, which makes us know, for the first time, the excellent work of Haydon, Levy, and Raynaud.

References

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, New York, NY, USA, 1983, Reissued by Dover Publications, New York, NY, USA, 2005.

T. X. Guo, The theory of probabilistic metric spaces with applications to random functional analysis [M.S. thesis], Xi'an Jiaotong University, Xi'an, China, 1989.

T. X. Guo, “Some basic theories of random normed linear spaces and random inner product spaces,” Acta Analysis Functionalis Applicata, vol. 1, no. 2, pp. 160–184, 1999.

T. X. Guo, “The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure,” Science in China. Series A, vol. 51, no. 9, pp. 1651–1663, 2008.

T. X. Guo, Random metric theory and its applications [Ph.D. thesis], Xi'an Jiaotong University, Xi'an, China, 1992.

R. Haydon, M. Levy, and Y. Raynaud, Randomly Normed Spaces, Hermann, Paris, France, 1991.

D. Filipović, M. Kupper, and N. Vogelpoth, “Separation and duality in locally L ⁰-convex modules,” Journal of Functional Analysis, vol. 256, no. 12, pp. 3996–4029, 2009.

T. X. Guo, “Relations between some basic results derived from two kinds of topologies for a random locally convex module,” Journal of Functional Analysis, vol. 258, no. 9, pp. 3024–3047, 2010.

T. X. Guo and Y. J. Yang, “Ekeland's variational principle for an -valued function on a complete random metric space,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 1–14, 2012.

S. E. Zhao and T. X. Guo, “The random subreflexivity of complete random normed modules,” International Journal of Mathematics, vol. 23, no. 3, Article ID 1250047, 14 pages, 2012.

T. X. Guo, “Recent progress in random metric theory and its applications to conditional risk measures,” Science China. Series A, vol. 54, no. 4, pp. 633–660, 2011.

T. X. Guo, S. E. Zhao, and X. L. Zeng, “On random convex analysis—the analytic foundation of the module approach to conditional risk measures,” http://arxiv.org/abs/1210.1848

N. Dunford and J. T. Schwartz, Linear Operators(I), Interscience, New York, NY, USA, 1957.

T. X. Guo, “Module homomorphisms on random normed modules,” Northeastern Mathematical Journal (China), vol. 12, no. 1, pp. 102–114, 1996.

J. P. LaSalle, “Pseudo-normed linear spaces,” Duke Mathematical Journal, vol. 8, pp. 131–135, 1941.

T. X. Guo, “Metrics, probabilistic metrics and random metrics,” Systems Science and Mathematical Sciences (China), vol. 8, no. 2, pp. 182–186, 1995.

Z. Y. You and L. H. Zhu, “The isometric metrization of probabilistic metric spaces,” Scientia Sinica, Series A, vol. 1, pp. 19–24, 1989.

K. Yosida, Functional Analysis, vol. 123, Beijing World Publishing, Beijing, China, 6th edition, 1999.

T. X. Guo, “Extension theorems of continuous random linear operators on random domains,” Journal of Mathematical Analysis and Applications, vol. 193, no. 1, pp. 15–27, 1995.

T. X. Guo and Z. Y. You, “The Riesz representation theorem for random inner product modules and its applications,” Chinese Annals of Mathematics. Series A, vol. 17, no. 3, pp. 361–364, 1996.

Z. Y. You and T. X. Guo, “Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in Lp (μ, X),” Journal of Approximation Theory, vol. 78, no. 3, pp. 314–320, 1994.

T. X. Guo and S. B. Li, “The James theorem in complete random normed modules,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 257–265, 2005.

T. X. Guo and Z. Y. You, “A note on pointwise best approximation,” Journal of Approximation Theory, vol. 93, no. 2, pp. 344–347, 1998.

T. X. Guo and X. Zhang, “Stone's representation theorem of a group of random unitary operators on complete complex random inner product modules,” Scientia Sinica, Series A, vol. 42, pp. 181–202, 2012.

T. X. Guo, “A characterization for a complete random normed module to be random reflexive,” Journal of Xiamen University. Natural Science, vol. 36, no. 4, pp. 167–170, 1997.

T. X. Guo, “Radon-Nikodým property of conjugate Banach spaces and w*-equivalence theorems of w*-measurable functions,” Science in China. Series A., vol. 39, no. 10, pp. 1034–1041, 1996.

T. X. Guo, “Representation theorems of the dual of Lebesgue-Bochner function spaces,” Science in China. Series A, vol. 43, no. 3, pp. 234–243, 2000.

T. X. Guo, H. Xiao, and X. Chen, “A basic strict separation theorem in random locally convex modules,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 9, pp. 3794–3804, 2009.

M. Z. Wu, “The Bishop-Phelps theorem in complete random normed modules endowed with the (ϵ, λ)-topology,” Journal of Mathematical Analysis and Applications, vol. 391, no. 2, pp. 648–652, 2012.