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

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, 𝐿^𝑞(𝑆^∗)≅(𝐿^𝑝(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,𝐿^𝑝(𝑆)(𝐿^𝑞(𝑆^∗)) the corresponding 𝐿^𝑝 (resp., 𝐿^𝑞) space derived from 𝑆 (resp., 𝑆^∗), and (𝐿^𝑝(𝑆))^′ the ordinary conjugate space of 𝐿^𝑝(𝑆).

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 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, (𝐿^𝑝(𝑆))^′≅𝐿^𝑞(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),  𝐿⁰(ℱ) the set of equivalence classes of extended real-valued ℱ-measurable functions on (Ω,ℱ,𝜇) and 𝐿⁰(ℱ)∶=𝐿⁰(ℱ,𝑅).

𝐿⁰(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 𝐿⁰(ℱ), respectively. It is well known from [13] that every subset 𝐻 in 𝐿⁰(ℱ) has a supremum and infimum, denoted by ∨𝐻 and ∧𝐻, respectively, and there are countable subsets {𝑎_𝑛,𝑛∈𝑁} and {𝑏_𝑛,𝑛∈𝑁} of 𝐻 such that ∨𝐻=∨_𝑛≥1𝑎_𝑛 and ∧𝐻=∧_𝑛≥1𝑏_𝑛. Furthermore, if, in addition, 𝐻 is upward directed or downward directed, then {𝑎_𝑛,𝑛∈𝑁} and {𝑏_𝑛,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: 𝐿⁰₊(ℱ)={𝜉∈𝐿⁰(ℱ)∣𝜉≥0},  𝐿⁰₊(ℱ)={𝜉∈𝐿⁰(ℱ)∣𝜉≥0},  𝐿⁰₊₊(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 𝐿⁰₊(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 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 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∈𝐿⁰₊(ℱ) such that ‖𝑇𝑥‖₁≤𝜉‖𝑥‖,forall𝑥∈𝑆, in which case ‖𝑇‖ is defined to be ∧{𝜉∈𝐿⁰₊(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 operator 𝑇^∗∶𝑆^∗₂→𝑆^∗₁ for an a.s. bounded linear operator 𝑇 from (𝑆₁,‖⋅‖₁) to (𝑆₂,‖⋅‖₂) as follows: (𝑇^∗𝑓)(𝑥)=𝑓(𝑇𝑥), for all 𝑓∈𝑆^∗₂,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 function 𝑇∶[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].

Although 𝑡-norms are widely used in the theory of probabilistic metric spaces, weak 𝑡-norms have their own advantages, for example, for a family {𝑇_𝛼,𝛼∈∧} of weak 𝑡-norms, 𝑇 defined by 𝑇(𝑎,𝑏)=sup{𝑇_𝛼(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∣lim_{𝑥→−∞}𝐹(𝑥)=0 and lim_𝑥→+∞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 𝑁_𝜉 is defined by 𝑁_𝜉(𝑡)=𝑃{𝜔∈Ω∣𝜉(𝜔)<𝑡}, for  all 𝑡∈[−∞,+∞)and𝑁_𝜉(+∞)=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) 𝑁_𝑥=𝜖₀ if and only if 𝑥=𝜃(the null element in 𝑆),(MPN-2) 𝑁_𝛼𝑥(𝑡)=𝑁_𝑥(𝑡/|𝛼|), for all 𝑡≥0,𝛼∈𝐾⧵{0} and 𝑥∈𝑆,(MPN-3) 𝑁_𝑥+𝑦(𝑢+𝑣)≥𝑇(𝑁_𝑥(𝑢),𝑁_𝑦(𝑣)), for all 𝑢,𝑣≥0 and 𝑥,𝑦∈𝑆. Here 𝑁_𝑥∶=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 {𝑝_𝛼}_𝛼∈∧ a family of mappings from 𝑆 to 𝑅⁺∶=[0,+∞) and indexed by a directed set ∧; then (𝑆,{𝑝_𝛼}_𝛼∈∧) is called a pseudonormed linear space if the following are satisfied:(PNS-1) 𝑝_𝛼(𝛽𝑥)=|𝛽|𝑝_𝛼(𝑥), for all 𝛽∈𝐾,𝑥∈𝑆, and 𝛼∈∧,(PNS-2) 𝑝_𝛼1(𝑥)≤𝑝_𝛼2(𝑥), for all 𝛼₁,𝛼₂∈∧ such that 𝛼₁≤𝛼₂,𝑥∈𝑆,(PNS-3) for any 𝛼∈∧, there exists 𝛼^′∈∧ such that 𝑝_𝛼(𝑥+𝑦)≤𝑝_𝛼′(𝑥)+𝑝_𝛼′(𝑦), for all 𝑥,𝑦∈𝑆.

Let (𝑆,{𝑝_𝛼}_𝛼∈∧) be a pseudonormed linear space. For any 𝜖>0 and 𝛼∈∧, let 𝑈_𝜃(𝛼,𝜖)={𝑥∈𝑆∣𝑝_𝛼(𝑥)<𝜖}. Then 𝒰_𝜃={𝑈_𝜃(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 {𝑝_𝛼}_𝛼∈∧. Conversely any linear topology for 𝑆 can be induced by some {𝑝_𝛼}_𝛼∈∧ such that (𝑆,{𝑝_𝛼}_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 𝑝_𝑟∶𝑆→[0,+∞) by 𝑝_𝑟(𝑥)=sup{𝑡≥0∣𝑁_𝑥(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)sup_0<𝑎<1𝑇(𝑎,𝑎)=1 if and only if (𝑆,{𝑝_𝑟}_𝑟∈(0,1)) is a pseudonormed linear space; namely, for each 𝑟∈(0,1) there exists 𝑠∈(0,1) such that 𝑝_𝑟(𝑥+𝑦)≤𝑝_𝑠(𝑥)+𝑝_𝑠(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 𝑝_𝑟 is a seminorm on 𝑆 for each 𝑟∈(0,1); namely, (𝑆,{𝑝_𝑟}_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 𝑝_𝑟=‖⋅‖, for all 𝑟∈(0,1).

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 (𝑆,𝒩,𝑇) be an M-PN space such that sup_0<𝑎<1𝑇(𝑎,𝑎)=1. For any positive numbers 𝜖 and 𝜆 with 0<𝜆<1, let 𝑈_𝜃(𝜖,𝜆)={𝑥∈𝑆∣𝑁_𝑥(𝜖)>1−𝜆}; then 𝒰_𝜃={𝑈_𝜃(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 Theorem 8, one can easily see that the (𝜖,𝜆)-topology for an M-PN space (𝑆,𝒩,𝑇) with sup_0<𝑎<1𝑇(𝑎,𝑎)=1 is exactly the one induced by the family {𝑝_𝑟}_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 sup_0<𝑎<1𝑇(𝑎,𝑎)=1 and the theory of pseudonormed linear space (𝑆,{𝑝_𝑟}_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 {𝑝_𝑟}_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 sup_0<𝑎<1𝑇(𝑎,𝑎)=1 and 𝐴 a subset of 𝑆.𝐷_𝐴∶[−∞,+∞]→[0,1] is defined by 𝐷_𝐴(𝑡)=sup_𝑟<𝑡(inf{𝑁_𝑥(𝑟)∶𝑥∈𝐴}), for all 𝑡∈(−∞,+∞),𝐷_𝐴(−∞)=0, and 𝐷_𝐴(+∞)=1, called the probabilistic diameter of 𝐴. If 𝐷_𝐴∈𝐷⁺, then 𝐴 is said to be probabilistically bounded.

Proposition 11 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 𝑁_𝑥(𝑡)=𝜇{𝜔∈Ω∣‖𝑥‖(𝜔)<𝑡},forall𝑡≥0 and 𝑥∈𝑆; namely, 𝑁_𝑥 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 𝑈_𝜃(𝐴,𝜖,𝜆)={𝑥∈𝑆∣𝜇{𝜔∈𝐴∣‖𝑥‖(𝜔)<𝜖}>𝜇(𝐴)−𝜆}. Then 𝒰_𝜃={𝑈_𝜃(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 ‖⋅‖.

Proposition 13 below is a straightforward verification by definition.

Proposition 13. Let (𝑆,‖⋅‖) be an RN space over 𝐾 with base (Ω,ℱ,𝜇) and {𝐴_𝑛∣𝑛∈𝑁} a countable partition of Ω to ℱ such that 0<𝜇(𝐴_𝑛)<+∞. Then one has the following.(1)A sequence {𝑥_𝑛𝑥∈𝑁} in 𝑆 converges in the (𝜖,𝜆)-topology to 𝑥 in 𝑆 if and only if {‖𝑥_𝑛−𝑥‖∣𝑛∈𝑁} converges to 0 locally in measure; namely, {‖𝑥_𝑛−𝑥‖∣𝑛∈𝑁} converges to 0 in measure 𝜇 on each 𝐴∈ℱ₊. (2)The (𝜖,𝜆)-topology for 𝑆 is exactly the linear topology induced by the quasinorm |‖⋅‖| defined by |‖𝑥‖|=∑^∞_𝑛=1(1/2^𝑛)∫_𝐴𝑛(‖𝑥‖)/(1+‖𝑥‖)𝑑𝜇 for all 𝑥∈𝑆. (3)Let 𝑃∶ℱ→[0,1] be defined by 𝑃(𝐴)=∑^∞_𝑛=1(1/2^𝑛)(𝜇(𝐴∩𝐴_𝑛)/𝜇(𝐴_𝑛)); then 𝑃 is a probability measure equivalent to 𝜇 and (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}∈𝐿⁰₊(ℱ).

In the sequel, the (𝜖,𝜆)-topology for every RN space is always denoted by 𝒯_𝜖,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 𝒯_𝜖,𝜆-bounded, at which time and when (Ω,ℱ,𝜇) is a probability space, 𝐷_𝐴=𝑁_𝜉, where 𝜉=∨{‖𝑎‖∶𝑎∈𝐴} and 𝑁_𝜉 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 {𝑥_𝑛,𝑛∈𝑁} in 𝐴 such that {‖𝑥_𝑛‖,𝑛∈𝑁} converges a.s. to 𝜉 in a nondecreasing manner. Let (𝑆,𝒩,𝑇) be the M-PN space determined by (𝑆,‖⋅‖); then {𝑁_𝑥𝑛,𝑛∈𝑁} converges weakly to 𝑁_𝜉; it is easy to check that 𝑁_𝜉=𝐷_{{𝑥𝑛,𝑛∈𝑁}} (namely, the probabilistic diameter of {𝑥_𝑛,𝑛∈𝑁}), and hence 𝑁_𝜉≥𝐷_𝐴. But 𝑁_𝜉≤𝐷_𝐴 is clear, then 𝑁_𝜉=𝐷_𝐴. Since 𝐴 is 𝒯_𝜉,𝜆-bounded, 𝐷_𝐴∈𝐷⁺, which shows that 𝜉∈𝐿⁰₊(ℱ).

Proposition 17 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_𝐷̃𝐴+𝐼̃_𝐷𝐶𝐴⊂𝐴 for any 𝐷∈ℱ, where 𝐷^𝐶=Ω⧵𝐷 and 𝐼_𝐷̃ 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_𝐷̃𝑎₂+𝐼̃_𝐷𝐶𝑎₁ is such that ‖𝑎₃‖=𝐼_𝐷̃‖𝑎₂‖+𝐼̃_𝐷𝐶‖𝑎₁‖=‖𝑎₁‖∨‖𝑎₂‖. Since 𝑎₃∈𝐴, the proof is complete.

It is easy to see that (𝐿⁰(ℱ,𝐾),𝒯_𝜖,𝜆) is a topological algebra over 𝐾 and (𝑆,𝒯_𝜖,𝜆) is a topological module over (𝐿⁰(ℱ,𝐾),𝒯_𝜖,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 𝒯_𝑐-open set if for each 𝑥∈𝐺 there exists some 𝜖∈𝐿⁰₊₊(ℱ) such that 𝑥+𝑈_𝜃(𝜖)⊂𝐺, where 𝑈_𝜃={𝑦∈𝑆∣‖𝑦‖≤𝜖}. Denote by 𝒯_𝑐 the family of 𝒯_𝑐-open sets; then 𝒯_𝑐 forms a Hausdorff topology for 𝑆, called the locally 𝐿⁰-convex topology induced by ‖⋅‖.

It is easy to check that the locally 𝐿⁰-convex topology is much stronger than the (𝜖,𝜆)-topology for a given RN module; (𝐿⁰(ℱ,𝐾),𝒯_𝑐) 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,𝒯_𝑐) is a topological module over the topological ring (𝐿⁰(ℱ,𝐾),𝒯_𝑐);compare [7]. From now on, the locally 𝐿⁰-convex topology for every RN module is always denoted by 𝒯_𝑐 when no confusion occurs.

Definition 19. Let 𝑆 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∈𝐿⁰₊(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∈𝐿⁰₊₊(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 𝒯_𝑐-bounded if 𝐴 is 𝐿⁰-absorbed by every 𝒯_𝑐-neighborhood of the null element.

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

For the sake of convenience, 𝐼_𝐴 always denotes the characteristic function of 𝐴∈ℱ and 𝐼_𝐴̃ the equivalence class of 𝐼_𝐴. 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_𝐴̃.

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 (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,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆); (2)𝑇 is a.s. bounded if and only if  𝑇 is a continuous module homomorphism from (𝑆₁,𝒯_𝑐) to (𝑆₂,𝒯_𝑐).

Proof

(1) Necessity. Since 𝑇 is a.s. bounded, 𝑇 must be continuous from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆); it remains to prove that 𝑇 is also a module homomorphism; it suffices to prove that 𝑇(𝐼_𝐴̃𝑥)=𝐼_𝐴̃𝑇(𝑥), for all 𝑥∈𝑆₁ and 𝐴∈ℱ, since 𝑇 is linear. Since ‖𝑇(𝐼_𝐴̃𝑥)‖₂≤‖𝑇‖⋅‖𝐼_𝐴̃𝑥‖₁=𝐼_𝐴̃‖𝑇‖⋅‖𝑥‖₁, 𝐼̃_𝐴𝑐⋅𝑇(𝐼_𝐴̃𝑥)=𝜃, for all 𝐴∈ℱ. Then 𝑇(𝐼_𝐴̃𝑥)=(𝐼_𝐴̃+𝐼̃_𝐴𝑐)⋅𝑇(𝐼_𝐴̃𝑥)=𝐼_𝐴̃⋅𝑇(𝐼_𝐴̃𝑥), for all 𝐴∈ℱ. On the other hand, 𝐼_𝐴̃𝑇(𝑥)=𝐼_𝐴̃⋅𝑇(𝐼_𝐴̃𝑥+𝐼̃_𝐴𝑐𝑥)=𝐼_𝐴̃⋅𝑇(𝐼_𝐴̃𝑥), for all 𝑥∈𝑆₁ and for all 𝐴∈ℱ. So, 𝐼_𝐴̃⋅𝑇(𝑥)=𝑇(𝐼_𝐴̃𝑥).

Sufficiency. 𝑆₁(1)∶={𝑥∈𝑆₁∣‖𝑥‖₁≤1} is a.s. bounded, and hence also 𝒯_𝜖,𝜆-bounded; further 𝑇(𝑆₁(1)) is 𝒯_𝜖,𝜆-bounded since 𝑇 is a continuous linear operator. Besides, 𝐼_𝐴̃⋅𝑇(𝑆₁(1))+𝐼̃_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)}∈𝐿⁰₊(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,𝒯_𝑐) to (𝑆₂,𝒯_𝑐).

Sufficiency. Since 𝑆₂(1)∶={𝑦∈𝑆₂∣‖𝑦‖≤1} is a 𝒯_𝑐-neighborhood of the null element of 𝑆₂ there exists some 𝜖∈𝐿⁰₊₊(ℱ) such that 𝑇(𝑈_𝜃(𝜖))⊂𝑆₂(1), where 𝑈_𝜃(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,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆) or from (𝑆₁,𝒯_𝑐) to (𝑆₂,𝒯_𝑐), the process of proof of Theorem 21 has implied that ‖𝑇‖=∨{‖𝑇(𝑥)‖₂∣𝑥∈𝑆₁(1)}; further we have 𝑁_𝑇=𝐷_{𝑇(𝑆1(1))} by Proposition 16, where 𝑁_𝑇 is the probabilistic norm of 𝑇, namely, the distribution function of ‖𝑇‖, and 𝐷_{𝑇(𝑆1(1))} is the probabilistic diameter of 𝑇(𝑆₁(1)).

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 (𝑆,‖⋅‖) be an RN module over 𝐾 with base (Ω,ℱ,𝜇) and 𝑓∶𝑆→𝐿⁰₊(ℱ) such that the following two conditions are satisfied: (1)𝑓(𝛼𝑥)=𝛼⋅𝑓(𝑥), for all 𝑥∈𝑆 and all nonnegative numbers 𝛼;(2)𝑓(𝑥+𝑦)≤𝑓(𝑥)+𝑓(𝑦), for all 𝑥,𝑦∈𝑆. Then 𝑓 is a.s. bounded; namely, there is some 𝜉∈𝐿⁰₊(ℱ) such that 𝑓(𝑥)≤𝜉‖𝑥‖, for all𝑥∈𝑆, if and only if 𝑓 is continuous from (𝑆,𝒯_𝜖,𝜆) to (𝐿⁰(ℱ),𝒯_𝜖,𝜆) and 𝑓(𝜂𝑥)=𝜂⋅𝑓(𝑥),forall𝑥∈𝑆 and 𝜂∈𝐿⁰₊(ℱ), at which time ‖𝑓‖=∨{𝑓(𝑥)∣𝑥∈𝑆(1)}, where ‖𝑓‖=∧{𝜉∈𝐿⁰₊(ℱ)∣𝑓(𝑥)≤𝜉‖𝑥‖,∀𝑥∈𝑆}; furthermore if, in addition, (Ω,ℱ,𝜇) is a probability space, then 𝑁_𝑓 (the distribution function of ‖𝑓‖)=𝐷_{𝑓(𝑆(1))}.

It is well known that 𝐵(𝑆₁,𝑆₂) is a Banach space when 𝑆₁ and 𝑆₂ are normed spaces and 𝑆₂ is complete. Similarly, 𝐵(𝑆₁,𝑆₂) is 𝒯_𝜖,𝜆-complete when (𝑆₁,‖⋅‖₁) and (𝑆₂,‖⋅‖₂) are RN modules and 𝑆₂ is 𝒯_𝜖,𝜆-complete, which is independently pointed out by Guo in [5, 14] and Haydon et al. in [6]; in particular 𝑆^∗ is 𝒯_𝜖,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 𝒯_𝜖,𝜆-complete; then 𝐵(𝑆₁,𝑆₂) is 𝒯_𝜖,𝜆-complete.

When 𝑆₁ 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 𝒯_𝜖,𝜆-closed submodule generated by 𝐽(𝑆) in 𝑆^∗∗ is called the 𝒯_𝜖,𝜆-closed submodule generated by 𝑆, denoted by 𝑀(𝑆); it is, obviously, a 𝒯_𝜖,𝜆-complete RN module.

Lemma 26 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 𝒯_𝜖,𝜆-complete. Then 𝐵(𝑆₁,𝐵₂) is isomorphic to a 𝒯_𝜖,𝜆-closed subspace of 𝐵(𝑀(𝑆₁,𝑀(𝑆₂)) in a random-norm-preserving way.

Proof. Define 𝐿∶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 𝒯_𝜖,𝜆-closed subspace of 𝐵(𝑀(𝑆₁),𝑀(𝑆₂)). Let {𝑇_𝑛,𝑛∈𝑁} be a sequence in 𝐵(𝑆₁,𝑆₂) such that {𝐿(𝑇_𝑛),𝑛∈𝑁} converges in the (𝜖,𝜆)-topology to some 𝑇∈𝐵(𝑀(𝑆₁),𝑀(𝑆₂)). Then {𝑇_𝑛,𝑛∈𝑁} is also 𝒯_𝜖,𝜆-Cauchy in 𝐵(𝑆₁,𝑆₂). We can, without loss of generality, assume that {‖𝑇_𝑛‖,𝑛∈𝑁} converges a.s. to some 𝜉. Define 𝑇∶𝑆₁→𝑆₂ by 𝑇(𝑥)=𝒯_𝜖,𝜆−lim_𝑛→∞𝑇_𝑛(𝑥),forall𝑥∈𝑆₁; then 𝑇 is well defined since 𝑆₂ is 𝒯_𝜖,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. Since 𝐵(𝑀(𝑆₁),𝑀(𝑆₂)) is always 𝒯_𝜖,𝜆-complete, so is 𝐵(𝑆₁,𝑆₂) when 𝑆₂ is 𝒯_𝜖,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]). Let 𝑋 be a linear topological space over 𝐾 of second category and (𝑌,|‖⋅‖|) a quasinormed linear space. Let {𝑇_𝛼,𝛼∈∧} be a family of continuous mappings from 𝑋 to 𝑌 such that the following three properties are satisfied:(1)|‖𝑇_𝛼(𝑥+𝑦)‖|≤|‖𝑇_𝛼(𝑥)‖|+|‖𝑇_𝛼(𝑦)‖|, for all 𝑥,𝑦∈𝑋 and 𝛼∈∧;(2)|‖𝑇_𝛼(𝑎𝑥)‖|=|‖𝑎𝑇_𝛼(𝑥)‖|, for all 𝑥∈𝑋,𝛼∈∧, and 𝑎≥0;(3){𝑇_𝛼(𝑥),𝛼∈∧} is bounded with respect to the linear topology induced by |‖⋅‖| for each 𝑥∈𝑋. Then lim_𝑥→𝜃𝑇_𝛼(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 𝒯_𝜖,𝜆-complete. Let {𝑇_𝛼∶𝛼∈∧}. be a family of continuous module homomorphisms from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆). Then, one has the following: (1){𝑇_𝛼∶𝛼∈∧} is 𝒯_𝜖,𝜆-bounded in 𝐵(𝑆₁,𝑆₂) if and only if {𝑇_𝛼(𝑥)∶𝛼∈∧} is 𝒯_𝜖,𝜆-bounded in 𝑆₂ for each 𝑥∈𝑆₁; (2){𝑇_𝛼∶𝛼∈∧} is a.s. bounded in 𝐵(𝑆₁,𝑆₂) if and only if {𝑇_𝛼(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 {𝑇_𝛼∶𝛼∈∧} be 𝒯_𝜖,𝜆-bounded in 𝐵(𝑆₁,𝑆₂); namely, 𝐷_{{𝑇𝛼∶𝛼∈∧}}∈𝐷⁺. For each 𝑥∈𝑆₁, since ‖𝑇_𝛼(𝑥)‖₂≤‖𝑇_𝛼‖‖𝑥‖₁,forall𝛼∈∧, 𝜇{𝜔∈Ω∣‖𝑇_𝛼(𝑥)‖₂(𝜔)<𝑡}≥𝜇({𝜔∈Ω∣‖𝑇_𝛼‖₂(𝜔)<2√𝑡}∩{𝜔∈Ω∣‖𝑥‖₁(𝜔)<(1/2)√𝑡})≥𝜇{𝜔∈Ω∣‖𝑇_𝛼‖₂(𝜔)<2√𝑡}+𝜇{𝜔∈Ω∣‖𝑥‖₁(𝜔)<(1/2)√𝑡}−1; namely, 𝑁_{𝑇𝛼(𝑥)}(𝑡)≥𝑁_𝑇𝛼(2√𝑡)+𝑁_𝑥((1/2)√𝑡)−1, for all 𝛼∈∧and𝑡>0. Then, 𝐷_{{𝑇𝛼(𝑥)∶𝛼∈∧}}(𝑡)≥𝑊(𝐷_{{𝑇𝛼∶𝛼∈∧}}(2√𝑡),𝑁_𝑥((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⁺and𝑁_𝑥∈𝐷⁺ for any 𝑥∈𝑆₁, then 𝐷_{{𝑇𝛼(𝑥)∶𝛼∈∧}}∈𝐷⁺; namely, {𝑇_𝛼(𝑥)∶𝛼∈∧} is 𝒯_𝜖,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,𝒯_𝜖,𝜆) is a linear topological space of the second category and it is also clear that {𝑇_𝛼∶𝛼∈∧} satisfies all the three conditions of Proposition 28, lim_𝑥→𝜃𝑇_𝛼(𝑥)=0 uniformly in 𝛼∈∧ by Proposition 28, which certainly implies that ⋃_𝛼∈∧𝑇_𝛼(𝐴) is 𝒯_𝜖,𝜆-bounded in 𝑆₂ for each 𝒯_𝜖,𝜆-bounded set 𝐴 in 𝑆₁, in particular ⋃_𝛼∈∧𝑇_𝛼(𝑆₁(1)) is 𝒯_𝜖,𝜆-bounded. By Remark 23, 𝑁_𝑇𝛼=𝐷_{𝑇𝛼(𝑆1(1))}, for all 𝛼∈∧. Define 𝐹∶[−∞,+∞]→[0,1] by 𝐹(𝑡)=inf_𝛼∈∧𝐷_{𝑇𝛼(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∈∧𝑇_𝛼(𝑆₁(1)) by 𝐴; then one can easily check that 𝐷_𝐴=𝑙⁻𝐹=𝐷_{{𝑇𝛼∶𝛼∈∧}}; then {𝑇_𝛼∶𝛼∈∧} is 𝒯_𝜖,𝜆-bounded since 𝐷_𝐴∈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 𝑓_𝐹∶𝑆₁→𝐿⁰₊(ℱ) by 𝑓_𝐹(𝑥)=∨{‖𝑇_𝛼𝑥‖₂∣𝛼∈𝐹}, for all 𝑥∈𝑆₁; then 𝑓_𝐹 is continuous from (𝑆₁,𝒯_𝜖,𝜆) to (𝐿⁰(ℱ,𝐾),𝒯_𝜖,𝜆) and 𝑓_𝐹(𝜉𝑥)=𝜉⋅𝑓_𝐹(𝑥), for all 𝜉∈𝐿⁰₊(ℱ) and 𝑥∈𝑆₁, and hence 𝑓_𝐹 is a.s. bounded and ‖𝑓_𝐹‖=∨{𝑓_𝐹(𝑥)∣𝑥∈𝑆₁(1)}=∨{‖𝑇_𝛼‖∶𝛼∈𝐹}. It is obvious that {‖𝑓_𝐹‖∣𝐹∈ℱ(∧)}=∨{‖𝑇_𝛼‖∣𝛼∈∧}, so we only need to prove that {‖𝑓_𝐹‖∣𝐹∈ℱ(∧)} is a.s. bounded, which is equivalent to the fact that {‖𝑓_𝐹‖∣𝐹∈ℱ(∧)} is 𝒯_𝜖,𝜆-bounded in 𝐿⁰₊(ℱ) by Proposition 16. Since ∨{𝑓_𝐹(𝑥)∣𝐹∈ℱ(∧)}=∨{‖𝑇_𝛼(𝑥)‖∣𝛼∈∧} for each 𝑥∈𝑆₁, {𝑓_𝐹(𝑥)∣𝐹∈ℱ(∧)} is a.s. bounded and hence also 𝒯_𝜖,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 {‖𝑓_𝐹‖∣𝐹∈ℱ(∧)} is 𝒯_𝜖,𝜆-bounded since {𝑓_𝐹∣𝐹∈ℱ(∧)} 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 𝒯_𝜖,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 𝒯_𝜖,𝜆-complete RN modules over 𝐾 with base (Ω,ℱ,𝜇) and 𝑇∶𝑆₁→𝑆₂ a module homomorphism. Then 𝑇 is continuous from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆) if and only if 𝑇 is 𝒯_𝜖,𝜆-closed (namely, the graph of 𝑇 is 𝒯_𝜖,𝜆-closed in 𝑆₁×𝑆₂).

Theorem 31. Let (𝑆₁,‖⋅‖₁) and (𝑆₂,‖⋅‖₂) be 𝒯_𝜖,𝜆-complete RN modules over 𝐾 with base (Ω,ℱ,𝜇) and 𝑇 a surjective continuous module homomorphism from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆). Then 𝑇 is 𝒯_𝜖,𝜆-open; namely, 𝑇(𝐺) is 𝒯_𝜖,𝜆-open for each 𝒯_𝜖,𝜆-open subset 𝐺 of 𝑆₁.

Theorem 32. Let (𝑆₁,‖⋅‖₁) and (𝑆₂,‖⋅‖₂) be 𝒯_𝜖,𝜆-complete RN modules over 𝐾 with base (Ω,ℱ,𝜇) and 𝑇 a bijective continuous module homomorphism from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆). Then 𝑇⁻¹ is also a continuous module homomorphism from (𝑆₂,𝒯_𝜖,𝜆) to (𝑆₁,𝒯_𝜖,𝜆).

To give the versions of Theorems 29 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 {𝐴_𝑛,𝑛∈𝑁} of Ω to ℱ such that 𝐼̃_𝐴𝑛𝑥=𝐼̃_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]). Let 𝑆 be an 𝐿⁰(ℱ,𝐾)-module. A subset 𝐺 of 𝑆 is said to have the countable concatenation property if for each sequence {𝑔_𝑛,𝑛∈𝑁} in 𝐺 and each countable partition {𝐴_𝑛,𝑛∈𝑁} of Ω to ℱ, there is 𝑔∈𝐺 such that 𝐼̃_𝐴𝑛𝑔=𝐼̃_𝐴𝑛𝑔_𝑛, 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 𝒯_𝜖,𝜆-complete if and only if it is 𝒯_𝑐-complete and 𝑆 has the countable concatenation property.

Theorem 36 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 𝒯_𝑐-complete and has the countable concatenation property. Let {𝑇_𝛼∶𝛼∈∧} be a family of continuous module homomorphism from (𝑆₁,𝒯_𝑐) to (𝑆₂,𝒯_𝑐); then {𝑇_𝛼∶𝛼∈∧} is 𝒯_𝑐-bounded if and only if {𝑇_𝛼(𝑥)∶𝛼∈∧} is 𝒯_𝑐-bounded for each 𝑥∈𝑆₁.

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

Theorem 37. Let (𝑆₁,‖⋅‖₁) and (𝑆₂,‖⋅‖₂) be two 𝒯_𝑐-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,𝒯_𝑐) to (𝑆₂,𝒯_𝑐) if and only if  𝑇 is 𝒯_𝑐-closed (namely, the graph of  𝑇 is 𝒯_𝑐-closed in 𝑆₁×𝑆₂).

Proof. It is clear that the graph of 𝑇 has the countable concatenation property. By Theorem 21, 𝑇 is continuous from (𝑆₁,𝒯_𝜖,𝜆) to (𝑆₂,𝒯_𝜖,𝜆) if and only if it is continuous from (𝑆₁,𝒯_𝑐) to (𝑆₂,𝒯_𝑐). So, the proof follows from Propositions 34 and 35 and Theorem 30.

Theorem 38. Let (𝑆₁,‖⋅‖₁) and (𝑆₂,‖⋅‖₂) be two 𝒯_𝑐-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,𝒯_𝑐) to (𝑆₂,𝒯_𝑐), then 𝑇⁻¹ is also continuous module homomorphism from (𝑆₂,𝒯_𝑐) to (𝑆₁,𝒯_𝑐).

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

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

Lemma 39. Let (𝑆,‖⋅‖) be a 𝒯_𝑐-complete RN module over 𝐾 with base (Ω,ℱ,𝜇) and 𝑀 a 𝒯_𝑐-closed submodule of 𝑆 such that both 𝑆 and 𝑀 have the countable concatenation property. Then, (𝑆/𝑀,‖⋅‖_𝑀) is still a 𝒯_𝑐-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→𝐿⁰₊(ℱ) is defined by ‖𝑥+𝑀‖_𝑀=∧{‖𝑦‖∣𝑦∈𝑥+𝑀}.

Proof. By Proposition 35 both 𝑆 and 𝑀 are 𝒯_𝜖,𝜆-complete; then (𝑆/𝑀,‖⋅‖_𝑀) is a 𝒯_𝜖,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 𝒯_𝑐-complete RN modules over 𝐾 with base (Ω,ℱ,𝜇) such that