On Random Topological Structures

Abstract

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.

1. Introduction

The purpose of this paper is to give a comprehensive text to the study of nonlinear random analysis such as the study of fixed point theory, approximation theory and stability of functional equations in random normed spaces. The notion of random normed space goes back to Šherstnev [1] and Hadžić [2–4] who were dulled from Menger [5], and Schweizer and Sklar [4] works. Some authors [6–11] considered some properties of probabilistic normed and metric spaces also fuzzy metric and normed spaces [12–21]. Fixed point theory [3, 22–26], approximation theory [27–31], and stability of functional equations [32–38] are studied at random normed space and its depended space that is, fuzzy normed space.

This paper is introduced as a survey of the latest and new results on the following topics.(i)Basic theory of triangular norms.(ii)Topological structure of random normed spaces.(iii)Random functional analysis.(iv)Relationship between random normed spaces and fuzzy normed spaces.

2. Triangular Norms

Triangular norms first appeared in the framework of probabilistic metric spaces in the work Menger [5]. It also turns out that this is a crucial operation in several fields. Triangular norms are an indispensable tool for the interpretation of the conjunction in fuzzy logics [39] and, subsequently, for the intersection of fuzzy sets [40]. They are, however, interesting mathematical objects themselves. We refer to some papers and books for further details (see [2, 4, 41–44]).

Definition 2.1. A triangular norm (shorter 𝑡-norm) is a binary operation on the unit interval [0,1], that is, a function 𝑇∶[0,1]×[0,1]→[0,1] such that for all 𝑎,𝑏,𝑐∈[0,1] the following four axioms are satisfied:
(T1) 𝑇(𝑎,𝑏)=𝑇(𝑏,𝑎) (commutativity);
(T2) 𝑇(𝑎,(𝑇(𝑏,𝑐)))=𝑇(𝑇(𝑎,𝑏),𝑐) (associativity);
(T3) 𝑇(𝑎,1)=𝑎 (boundary condition);
(T4) 𝑇(𝑎,𝑏)≤𝑇(𝑎,𝑐) whenever 𝑏≤𝑐 (monotonicity).

The commutativity of (T1), boundary condition (T3), and the the monotonicity (T4) imply that for each 𝑡-norm 𝑇 and for each 𝑥∈[0,1] the following boundary conditions are also satisfied: 𝑇()()𝑇()()𝑥,1=𝑇1,𝑥=𝑥,𝑥,0=𝑇0,𝑥=0, (2.1) and therefore all 𝑡-norms coincide on the boundary of the unit square [0,1]².

The monotonicity of a 𝑡-norm 𝑇 in its second component (T3) is, together with the commutativity (T1), equivalent to the (joint) monotonicity in both components, that is, to 𝑇𝑥₁,𝑦₁≤𝑇𝑥₂,𝑦₂whenever𝑥₁≤𝑥₂,𝑦₁≤𝑦₂. (2.2) Basic examples are the Łukasiewicz 𝑡-norm 𝑇_𝐿: 𝑇_𝐿(𝑎,𝑏)=max(𝑎+𝑏−1,0),∀𝑎,𝑏∈[0,1] (2.3) and the 𝑡-norms 𝑇_𝑃,𝑇_𝑀,𝑇_𝐷, where 𝑇_𝑃()𝑇𝑎,𝑏∶=𝑎𝑏,_𝑀(){},𝑇𝑎,𝑏∶=min𝑎,𝑏_𝐷()()()𝑎,𝑏∶=min𝑎,𝑏,ifmaxa,b=1,0,otherwise. (2.4)

If, for two 𝑡-norms 𝑇₁ and 𝑇₂, the inequality 𝑇₁(𝑥,𝑦)≤𝑇₂(𝑥,𝑦) holds for all (𝑥,𝑦)∈[0,1]², then we say that 𝑇₁ is weaker than 𝑇₂ or, equivalently, that 𝑇₂ is stronger than 𝑇₂.

As a result of (2.2) we obtain, for each (𝑥,𝑦)∈[0,1]², 𝑇()𝑇𝑥,𝑦≤𝑇𝑥,1=𝑥,𝑥,𝑦≤𝑇1,𝑦=𝑦. (2.5) Since, for all (𝑥,𝑦)∈(0,1)², trivially 𝑇(𝑥,𝑦)≥0=𝑇_𝐷(𝑥,𝑦), we get for an arbitrary 𝑡-norm 𝑇, 𝑇_𝐷≤𝑇≤𝑇_𝑀. (2.6) That is, 𝑇_𝐷 is weaker and 𝑇_𝑀 is stronger than any other 𝑡-norms. Also since 𝑇_𝐿<𝑇_𝑃, we obtain the following ordering for four basic 𝑡-norms 𝑇_𝐷<𝑇_𝐿<𝑇_𝑃<𝑇_𝑀. (2.7)

Proposition 2.2 (see [2]). (i) The minimum 𝑇_𝑀 is the only 𝑡-norm satisfying 𝑇(𝑥,𝑥)=𝑥 for all 𝑥∈(0,1).
(ii) The weakest 𝑡-norm 𝑇_𝐷 is the only 𝑡-norm satisfying 𝑇(𝑥,𝑥)=0for all 𝑥∈(0,1).

Proposition 2.3 (see [2]). A 𝑡-norm 𝑇 is continuous if and only if it is continuous in its first component, that is, if for each 𝑦∈[0,1] the one-place function 𝑇⋅,𝑦∶[0,1]⟶[0,1],𝑥⟼𝑇𝑥,𝑦, (2.8) is continuous.

For example, the minimum 𝑇_𝑀 and Łukasiewicz 𝑡-norm 𝑇_𝐿 are continuous but the t-norm 𝑇^Δ defined by 𝑇^Δ𝑥,𝑦∶=𝑥𝑦2,ifmax𝑥,𝑦<1,𝑥𝑦,otherwise, (2.9) for 𝑥,𝑦∈[0,1], is not continuous.

Definition 2.4. (i) A 𝑡-norm 𝑇 is said to be strictly monotone if 𝑇𝑥,𝑦<𝑇(𝑥,𝑧)whenever𝑥∈(0,1),𝑦<𝑧. (2.10)(ii) A 𝑡-norm 𝑇 is said to be strict if it is continuous and strictly monotone.

For example, the 𝑡-norm 𝑇^Δ is strictly monotone but the minimum 𝑇_𝑀 and Łukasiewicz 𝑡-norm 𝑇_𝐿 are not.

Proposition 2.5 (see [2]). A 𝑡-norm 𝑇 is strictly monotone if and only if 𝑇𝑥,𝑦=𝑇(𝑥,𝑧),𝑥>0⟹𝑦=𝑧. (2.11)

If 𝑇 is a t-norm, then 𝑥_𝑇^(𝑛) is defined for every 𝑥∈[0,1] and 𝑛∈𝑁∪{0} by 1, if 𝑛=0 and 𝑇(𝑥_𝑇^(𝑛−1),𝑥), if 𝑛≥1.

Definition 2.6. A 𝑡-norm 𝑇 is said to be Archimedean if for all (𝑥,𝑦)∈(0,1)² there is an integer 𝑛∈ℕ such that 𝑥_𝑇^(𝑛)<𝑦. (2.12)

Proposition 2.7 (see [2]). A 𝑡-norm 𝑇 is Archimedean if and only if for each 𝑥∈(0,1) one has lim_𝑛→∞x_𝑇^(𝑛)=0. (2.13)

Proposition 2.8 (see [2]). If 𝑡-norm 𝑇 is Archimedean, then for each 𝑥∈(0,1) one has 𝑇(𝑥,𝑥)<𝑥. (2.14)

For example, the product 𝑇_𝑃, Łukasiewicz 𝑡-norm 𝑇_𝐿, and weakest 𝑡-norm 𝑇_𝐷 are all Archimedean but the minimum 𝑇_𝑀 is not an Archimedean 𝑡-norm.

A t-norm 𝑇 is said to be of Hadžić-type (we denote by 𝑇∈ℋ) if the family (𝑥_𝑇^(𝑛))_𝑛∈𝑁 is equicontinuous at 𝑥=1, that is, ∀𝜀∈(0,1)∃𝛿∈(0,1),𝑥>1−𝛿⟹𝑥_𝑇^(𝑛)>1−𝜀,∀𝑛≥1. (2.15)𝑇_𝑀 is a trivial example of a t-norm of Hadžić type, but 𝑇_𝑃 is not of Hadžić type.

Proposition 2.9 (see [2]). If a continuous 𝑡-norm 𝑇 is Archimedean, then it cannot be a 𝑡-norm of Hadžić type.

Other important triangular norms are [see [45]]:(i)the Sugeno-Weber family {𝑇^SW_𝜆}_{𝜆∈[−1,∞]} is defined by 𝑇^SW₋₁=𝑇_𝐷,𝑇^SW_∞=𝑇_𝑃 and 𝑇^SW_𝜆𝑥,𝑦=max0,𝑥+𝑦−1+𝜆𝑥𝑦1+𝜆 (2.16)if 𝜆∈(−1,∞);(ii)the Domby family {𝑇^𝐷_𝜆}_{𝜆∈[0,∞]}, defined by 𝑇_𝐷, if 𝜆=0,𝑇_𝑀, if 𝜆=∞ and 𝑇^𝐷_𝜆=1𝑥,𝑦1+((1−𝑥)/𝑥)^𝜆+1−𝑦/𝑦^𝜆1/𝜆 (2.17)if 𝜆∈(0,∞);(iii)the Aczel-Alsina family {𝑇^AA_𝜆}_{𝜆∈[0,∞]}, defined by 𝑇_𝐷, if 𝜆=0,𝑇_𝑀, if 𝜆=∞ and 𝑇^AA_𝜆𝑥,𝑦=𝑒^{−(|log𝑥|𝜆+|log𝑦|𝜆)1/𝜆} (2.18)

if 𝜆∈(0,∞).

A 𝑡-norm 𝑇 can be extended (by associativity) in a unique way to an 𝑛-array operation taking for any (𝑥₁,…,𝑥_𝑛)∈[0,1]^𝑛 the value 𝑇(𝑥₁,…,𝑥_𝑛) defined by T⁰_𝑖=1𝑥_𝑖=1,T^𝑛_𝑖=1𝑥_𝑖=𝑇T^𝑛−1_𝑖=1𝑥_𝑖,𝑥_𝑛=𝑇𝑥₁,…,𝑥_𝑛. (2.19)

𝑇 can also be extended to a countable operation taking for any sequence (𝑥_𝑛)_𝑛∈𝑁 in [0,1] the value T^∞_𝑖=1𝑥_𝑖=lim_𝑛→∞T^𝑛_𝑖=1𝑥_𝑖. (2.20) The limit on the right side of (2.20) exists since the sequence (T^𝑛_𝑖=1𝑥_𝑖)_𝑛∈ℕ is nonincreasing and bounded from below.

Proposition 2.10 (see [45]). (i) For 𝑇≥𝑇_𝐿, the following implication holds: lim_𝑛→∞T^∞_𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛<∞. (2.21)
(ii) If 𝑇 is of Hadžić-type, then lim_𝑛→∞T^∞_𝑖=1𝑥_𝑛+𝑖=1 (2.22) for every sequence (𝑥_𝑛)_𝑛∈𝑁 in [0,1] such that lim_𝑛→∞𝑥_𝑛=1.
(iii) If 𝑇∈{𝑇^AA_𝜆}_{𝜆∈(0,∞)}∪{𝑇^𝐷_𝜆}_{𝜆∈(0,∞)}, then lim_𝑛→∞T^∞_𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛^𝜆<∞. (2.23)
(iv) If 𝑇∈{𝑇^SW_𝜆}_{𝜆∈[−1,∞)}, then lim_𝑛→∞T^∞_𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛<∞. (2.24)

Definition 2.11. Let 𝑇 and 𝑇^′ be two continuous 𝑡-norms. Then 𝑇^′ dominates 𝑇, denoted by 𝑇^′≫𝑇, if for all 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈[0,1], 𝑇𝑇^′𝑥₁,𝑥₂,𝑇^′𝑦₁,𝑦₂≤𝑇^′𝑇𝑥₁,𝑦₁,𝑇𝑥₂,𝑦₂. (2.25)

Now, we extended definitions and results on the triangular norm on lattices. Let ℒ=(𝐿,≤_𝐿) be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and 0_ℒ=inf𝐿, 1_ℒ=sup𝐿.

Definition 2.12 (see [46]). A 𝑡-norm on 𝐿 is a mapping 𝒯∶(𝐿)²→𝐿 satisfying the following conditions:
(a) (∀𝑥∈𝐿)(𝒯(𝑥,1_ℒ)=𝑥) (boundary condition);
(b) (∀(𝑥,𝑦)∈(𝐿)²)(𝒯(𝑥,𝑦)=𝒯(𝑦,𝑥)) (commutativity);
(c) (∀(𝑥,𝑦,𝑧)∈(𝐿)³)(𝒯(𝑥,𝒯(𝑦,𝑧))=𝒯(𝒯(𝑥,𝑦),𝑧)) (associativity);
(d) (∀(𝑥,𝑥^′,𝑦,𝑦^′)∈(𝐿)⁴)(𝑥≤_𝐿𝑥^′ and 𝑦≤_𝐿𝑦^′⇒𝒯(𝑥,𝑦)≤_𝐿𝒯(𝑥^′,𝑦^′)) (monotonicity).

Let {𝑥_𝑛} be a sequence in 𝐿 converges to 𝑥∈𝐿 (equipped order topology). The 𝑡-norm 𝒯 is said to be a continuous 𝑡-norm if lim_𝑛→∞𝒯𝑥_𝑛,𝑦=𝒯𝑥,𝑦, (2.26) for each 𝑦∈𝐿.

Note that we put 𝒯=𝑇 whenever 𝐿=[0,1].

Definition 2.13 (see [46]). A continuous 𝑡-norm 𝒯 on 𝐿=[0,1]² is said to be continuous 𝑡-representable if there exist a continuous 𝑡-norm ∗ and a continuous 𝑡-conorm ⋄ on [0,1] such that, for all 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿, 𝒯𝑥,𝑦=𝑥₁∗𝑦₁,𝑥₂⋄𝑦₂. (2.27)

For example, 𝒯(𝑎,𝑏)=𝑎₁𝑏₁,min𝑎₂+𝑏₂,𝐌()=𝑎,1𝑎,𝑏min₁,𝑏₁,max𝑎₂,𝑏₂ (2.28) for all 𝑎=(𝑎₁,𝑎₂),𝑏=(𝑏₁,𝑏₂)∈[0,1]² are continuous 𝑡–representable.

Define the mapping 𝒯_∧ from 𝐿² to 𝐿 by: 𝒯_∧𝑥,𝑦=𝑥,if𝑦≥_𝐿𝑥,𝑦,if𝑥≥_𝐿𝑦. (2.29)

A negation on ℒ is any decreasing mapping 𝒩∶𝐿→𝐿 satisfying 𝒩(0_ℒ)=1_ℒ and 𝒩(1_ℒ)=0_ℒ. If 𝒩(𝒩(𝑥))=𝑥 for all 𝑥∈𝐿, then 𝒩 is called an involutive negation. In the following ℒ is endowed with a (fixed) negation 𝒩.

3. Distribution Functions and Fuzzy Sets

We denote, Δ⁺, the space of all distribution functions, that is, the space of all mappings 𝐹∶ℝ∪{−∞,+∞}→[0,1], such that 𝐹 is left-continuous and nondecreasing on ℝ, 𝐹(0)=0 and 𝐹(+∞)=1. 𝐷⁺ is a subset of Δ⁺ consisting of all functions 𝐹∈Δ⁺ for which 𝑙⁻𝐹(+∞)=1, where 𝑙⁻𝑓(𝑥) denotes the left limit of the function 𝑓 at the point 𝑥, that is, 𝑙⁻𝑓(𝑥)=lim_{𝑡→𝑥−}𝑓(𝑡). The space Δ⁺ is partially ordered by the usual point-wise ordering of functions, that is, 𝐹≤𝐺 if and only if 𝐹(𝑡)≤𝐺(𝑡) for all 𝑡 in ℝ. The maximal element for Δ⁺ in this order is the distribution function 𝜀₀ given by 𝜀₀(𝑡)=0,if𝑡≤0,1,if𝑡>0, (3.1)

Example 3.1. The function 𝐺(𝑡), defined by 𝐺(𝑡)=0,if𝑡≤0,1−𝑒^−𝑡,if𝑡>0, (3.2) is a distribution function. Since lim_𝑡→∞𝐺(𝑡)=1, then 𝐺∈𝐷⁺. Note that 𝐺(𝑡+𝑠)≥𝑇_𝑃(𝐺(𝑡),𝐺(𝑠)) for each 𝑡,𝑠>0.

Example 3.2. The function 𝐹(𝑡), defined by 𝐹(𝑡)=0if𝑡≤0,𝑡if0≤𝑡≤1,1if1≤𝑡, (3.3) is a distribution function. Since lim_𝑡→∞𝐹(𝑡)=1, then 𝐹∈𝐷⁺. Note that 𝐹(𝑡+𝑠)≥𝑇_𝑀(𝐹(𝑡),𝐹(𝑠)) for each 𝑡,𝑠>0.

Example 3.3 (see [6]). The function 𝐺_𝑝(𝑡), 𝑝>0, defined by 𝐺_𝑝(𝑡)=−𝑝0if𝑡≤0,exp^1/2if0<𝑡<+∞,1if𝑡=+∞, (3.4) is a distribution function. Since lim_𝑡→∞𝐺_𝑝(𝑡)≠1, then 𝐺∈Δ⁺⧵𝐷⁺. Note that 𝐺_𝑝(𝑡+𝑠)≥𝑇_𝑀(𝐺_𝑝(𝑡),𝐺_𝑝(𝑠)) for each 𝑡,𝑠>0.

Definition 3.4. If 𝑊 is a collection of objects denoted generically by 𝑤, then a fuzzy set 𝐴 in 𝑊 is a set of ordered pairs: 𝐴=𝑤,𝜆_𝐴(𝑤)∶𝑤∈𝑊. (3.5)𝜆_𝐴(𝑤) is called the membership function or grade of membership of 𝑤 in 𝐴 which maps 𝑊 to the membership space 𝑀. Note that, when 𝑀 contains only the two points 0 and 1, 𝐴 is nonfuzzy and 𝜆_𝐴(𝑤) is identical to the characteristic function of a non-fuzzy set. The range of the membership function is [0,1] or a complete lattice.

Example 3.5. Consider the fuzzy set 𝐴 which is real numbers considerably larger than 10, 𝐴=𝑤,𝜆_𝐴(𝑤)∶𝑤∈𝑊, (3.6) where 𝜆_𝐴(𝑤)=10,if𝑤<10,1+(𝑤−10)⁻²,if𝑤≥10. (3.7)

Example 3.6. Consider the fuzzy set 𝐴 which is real numbers close to 10, like (3.6) where 𝜆_𝐴(𝑤)=11+(𝑤−10)². (3.8)
Note that, in this paper, in short, we apply membership functions instead of fuzzy sets.

Definition 3.7 (see [47]). Let ℒ=(𝐿,≤_𝐿) be a complete lattice and let 𝑈 be a nonempty set called the universe. An ℒ- fuzzy set in U is defined as a mapping 𝒜∶𝑈→𝐿. For each 𝑢 in 𝑈, 𝒜(𝑢) represents the degree (in 𝐿) to which 𝑢 is an element of 𝒜.

Lemma 3.8 (see [46]). Consider the set 𝐿^∗ and operation ≤_𝐿^∗ defined by 𝐿^∗=𝑥₁,𝑥₂∶𝑥₁,𝑥₂∈[0,1]²and𝑥₁+𝑥₂,𝑥≤1₁,𝑥₂≤_𝐿^∗𝑦₁,𝑦₂⟺𝑥₁≤𝑦₁,𝑥₂≥𝑦₂ (3.9) for all (𝑥₁,𝑥₂),(𝑦₁,𝑦₂)∈𝐿^∗. Then (𝐿^∗,≤_𝐿^∗) is a complete lattice.

Definition 3.9 (see [48]). An intuitionistic fuzzy set 𝒜_𝜁,𝜂 in the universe 𝑈 is an object 𝒜_𝜁,𝜂={(𝑢,𝜁_𝒜(𝑢),𝜂_𝒜(𝑢))∶𝑢∈𝑈}, where 𝜁_𝒜(𝑢)∈[0,1] and 𝜂_𝒜(𝑢)∈[0,1] for all 𝑢∈𝑈 are called the membership degree and the nonmembership degree, respectively, of 𝑢 in 𝒜_𝜁,𝜂 and, furthermore, satisfy 𝜁_𝒜(𝑢)+𝜂_𝒜(𝑢)≤1.

Example 3.10. Consider the intuitionistic fuzzy set 𝒜_𝜁,𝜂 which is real numbers considerably larger than 10 for first place and real numbers close to 10 in second place, 𝒜_𝜁,𝜂=𝑤,𝜁_𝐴(𝑤),𝜂_𝐴(𝑤)∶𝑤∈𝑊, (3.10) where 𝜁_𝐴(𝑤),𝜂_𝐴(𝑤=1)0,1+(𝑤−10)²1,if𝑤<10,1+(𝑤−10)⁻²,11+(𝑤−10)²,if𝑤≥10. (3.11)
As we said above, we apply 𝒜_𝜁,𝜂(𝑤)=(𝜁_𝐴(𝑤),𝜂_𝐴(𝑤)).

4. Random Normed Spaces

Random (probabilistic) normed spaces were introduced by Šerstnev in 1962 [1] by means of a definition that was closely modelled on the theory of (classical) normed spaces and used to study the problem of best approximation in statistics. In the sequel, we will adopt usual terminology, notation, and conventions of the theory of random normed spaces, as in [4, 6–8].

Definition 4.1. A Menger probabilistic metric space (or random metric spaces) is a triple (𝑋,ℱ,𝑇), where 𝑋 is a nonempty set, 𝑇 is a continuous 𝑡-norm, and ℱ is a mapping from 𝑋×𝑋 into 𝐷⁺ such that, if 𝐹_𝑥,𝑦 denotes the value of ℱ at the pair (𝑥,𝑦), the following conditions hold:
(PM1) 𝐹_𝑥,𝑦(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=𝑦;
(PM2) 𝐹_𝑥,𝑦(𝑡)=𝐹_𝑦,𝑥(𝑡);
(PM3) 𝐹_𝑥,𝑧(𝑡+𝑠)≥𝑇(𝐹_𝑥,𝑦(𝑡),𝐹_𝑦,𝑧(𝑠)) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠≥0.

Definition 4.2 (see [1]). A random normed space (briefly, RN-space) is a triple (𝑋,𝜇,𝑇), where 𝑋 is a vector space, 𝑇 is a continuous 𝑡-norm, and 𝜇 is a mapping from 𝑋 into 𝐷⁺ such that the following conditions hold:
(RN1) 𝜇_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0;
(RN2) 𝜇_𝛼𝑥(𝑡)=𝜇_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝛼≠0;
(RN3) 𝜇_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(s)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

Example 4.3. Let (𝑋,‖⋅‖) be a linear normed space. Define 𝜇_𝑥(𝑡)=𝑡0,if𝑡≤0,𝑡+‖𝑥‖,if𝑡>0. (4.1) Then (𝑋,𝜇,𝑇_𝑃) is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). 𝑇_𝑃𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)=𝑡‖𝑥‖⋅𝑠𝑡+𝑦=1𝑠+𝑥‖)⋅11+(‖/𝑡𝑦≤11+/𝑠𝑥‖/(⋅11+(‖𝑡+𝑠))𝑦/()≤11+𝑡+𝑠‖𝑥‖+𝑦/()≤11+𝑡+𝑠/()=1+𝑥+𝑦𝑡+𝑠𝑡+𝑠𝑡+𝑠+𝑥+𝑦=𝜇_𝑥+𝑦(𝑡+𝑠) (4.2) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0. Also, (𝑋,𝜇,𝑇_𝑀) is a random normed space.

Example 4.4. Let (𝑋,‖⋅‖) be a linear normed spaces. Define 𝜇_𝑥(𝑡)=𝑒0,if𝑡≤0,_{−(‖𝑥‖/𝑡)},if𝑡>0. (4.3) Then (𝑋,𝜇,𝑇_𝑃) is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). 𝑇_𝑃𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)=𝑒^{−(‖𝑥‖/𝑡)}⋅𝑒^{−(‖𝑦‖/𝑠)}≤𝑒^{−(‖𝑥‖/(𝑡+𝑠))}⋅𝑒^{−(‖𝑦‖/(𝑡+𝑠))}=𝑒^{−((‖𝑥‖+‖𝑦‖)/(𝑡+𝑠))}≤𝑒^{−((‖𝑥+𝑦‖)/(𝑡+𝑠))}=𝜇_𝑥+𝑦(𝑡+𝑠) (4.4) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0. Also, (𝑋,𝜇,𝑇_𝑀) is a random normed space.

Example 4.5 (see [36]). Let (𝑋,‖⋅‖) be a linear normed space. Define 𝜇_𝑥(𝑡)=‖𝑥‖max1−𝑡,0,if𝑡>0,0,if𝑡≤0. (4.5) Then (𝑋,𝜇,𝑇_𝐿) is an RN-space (this was essentially proved by Mušthari in [49], see also [50]). Indeed, 𝜇_𝑥(𝑡)=1 for all 𝑡>0⇒‖𝑥‖/𝑡=0 for all 𝑡>0⇒𝑥=0 and obviously 𝜇_𝜆𝑥(𝑡)=𝜇_𝑥(𝑡/𝜆)forall𝑥∈𝑋 and 𝑡>0. Next, for every 𝑥,𝑦∈𝑋 and 𝑡,𝑠>0 we have 𝜇_𝑥+𝑦(𝑡+𝑠)=max1−𝑥+𝑦𝑡+𝑠,0=max1−𝑥+𝑦𝑥𝑡+𝑠,0=max1−+𝑦𝑡+𝑠𝑥𝑡+𝑠,0≥max1−𝑡−𝑦𝑠,0=𝑇_𝐿𝜇_𝑥(𝑡),𝜇_𝑦(𝑠). (4.6)

Let 𝜑 be a function defined on the real field ℝ into itself with the following properties:(a)𝜑(−𝑡)=𝜑(𝑡) for every 𝑡∈ℝ;(b)𝜑(1)=1;(c)𝜑 is strictly increasing and continuous on [0,∞), 𝜑(0)=0 and lim_𝛼→∞𝜑(𝛼)=∞.

Examples of such functions are 𝜑(𝑡)=|𝑡|; 𝜑(𝑡)=|𝑡|^𝑝, 𝑝∈(0,∞); 𝜑(𝑡)=2𝑡^2𝑛/(|𝑡|+1), 𝑛∈ℕ.

Definition 4.6 (see [51]). A random 𝜑-normed space is a triple (𝑋,𝜈,𝑇), where 𝑋 is a real vector space, 𝑇 is a continuous 𝑡–norm, and 𝜈 is a mapping from 𝑋 into 𝐷⁺ such that the following conditions hold:
(𝜑-RN1) 𝜈_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0;
(𝜑-RN2) 𝜈_𝛼𝑥(𝑡)=𝜈_𝑥(𝑡/𝜑(𝛼)) for all 𝑥 in 𝑋, 𝛼≠0 and 𝑡>0;
(𝜑-RN3) 𝜈_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜈_𝑥(𝑡),𝜈_y(𝑠)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

Example 4.7 (see [37]). An important example is the space (𝑋,𝜈,𝑇_𝑀), where (𝑋,‖⋅‖^𝑝) is a p-normed space and 𝜈_𝑥(𝑡)=𝑡0,if𝑡≤0,𝑡+‖𝑥‖^𝑝,𝑝∈(0,1],if𝑡>0. (4.7) (𝜑-RN1) and (𝜑-RN2) are obvious and we show (𝜑-RN3). Let 𝜈_𝑥(𝑡)≤𝜈_𝑦(𝑠), then for each 𝑥,𝑦∈𝑋 we have 𝑦^𝑝𝑠≤‖𝑥‖^𝑝𝑡. (4.8) Now, if 𝑥=𝑦, then we have 𝑡≤𝑠. Now, since we have ‖𝑥‖^𝑝𝑡+‖𝑥‖^𝑝𝑡≥‖𝑥‖^𝑝𝑡+𝑦^𝑝𝑠‖𝑥‖≥2^𝑝𝑦𝑡+𝑠+2^𝑝𝑡+𝑠≥2𝑥+𝑦^𝑝,𝑡+𝑠 (4.9) then it follows that 1+‖𝑥‖^𝑝𝑡≥1+‖𝑥+𝑦‖^𝑝𝑡+𝑠, (4.10) which implies that 𝜈_𝑥(𝑡)≤𝜈_𝑥+𝑦(𝑡+𝑠). Hence 𝜈_𝑥+𝑦(𝑡+𝑠)≥𝑇_𝑀(𝜈_𝑥(𝑡),𝜈_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

By a non-Archimedean field we mean a field 𝒦 equipped with a function (valuation) |⋅| from 𝒦 into [0,∞) such that |𝑟|=0 if and only if 𝑟=0, |𝑟𝑠|=|𝑟||𝑠|, and |𝑟+𝑠|≤max{|𝑟|,|𝑠|} for all 𝑟,𝑠∈𝒦. Clearly |1|=|−1|=1 and |𝑛|≤1 for all 𝑛∈ℕ. By the trivial valuation we mean the mapping |⋅| taking everything but 0 into 1 and |0|=0. Let 𝑋 be a vector space over a field 𝒦 with a non-Archimedean non-trivial valuation |⋅|, that is, that there is an 𝑎₀∈𝒦 such that |𝑎₀| is not in {0,1}.

The most important examples of non-Archimedean spaces are 𝑝-adic numbers. In 1897, Hensel [52] discovered the 𝑝-adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number 𝑝. For any nonzero rational number 𝑥, there exists a unique integer 𝑛_𝑥∈ℤ such that 𝑥=(𝑎/𝑏)𝑝^𝑛𝑥, where 𝑎 and 𝑏 are integers not divisible by 𝑝. Then |𝑥|_𝑝∶=𝑝^−𝑛𝑥 defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|_𝑝 is denoted by ℚ_𝑝, which is called the 𝑝-adic number field.

A function ‖⋅‖∶𝑋→[0,∞) is called a non-Archimedean norm if it satisfies the following conditions:(i)‖𝑥‖=0 if and only if 𝑥=0;(ii)for any 𝑟∈𝒦,𝑥∈𝑋, ‖𝑟𝑥‖=|𝑟|‖𝑥‖;(iii)the strong triangle inequality (ultrametric); namely, 𝑥+𝑦≤max‖𝑥‖,𝑦,∀𝑥,𝑦∈𝑋. (4.11)

Then (𝑋,‖⋅‖) is called a non-Archimedean normed space. Due to the fact that 𝑥_𝑛−𝑥_𝑚≤max𝑥_𝑗+1−𝑥_𝑗∶𝑚≤𝑗≤𝑛−1,∀𝑛,𝑚∈ℕ(𝑛>𝑚), (4.12) a sequence {𝑥_𝑛} is a Cauchy sequence if and only if {𝑥_𝑛+1−𝑥_𝑛} converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.

Definition 4.8. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple (𝑋,𝜇,𝑇), where 𝑋 is a linear space over a non-Archimedean field 𝒦, 𝑇 is a continuous 𝑡-norm, and 𝜇 is a mapping from 𝑋 into 𝐷⁺ such that the following conditions hold:
(NA-RN1) 𝜇_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0;
(NA-RN2) 𝜇_𝛼𝑥(𝑡)=𝜇_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝑡>0,𝛼≠0;
(NA-RN3) 𝜇_𝑥+𝑦(max{𝑡,𝑠})≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠≥0.
It is easy to see that, if (NA-RN3) holds then so will (RN3) 𝜇_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)).

Example 4.9. As a classical example, if (𝑋,‖⋅‖) is a non-Archimedean normed linear space, then the triple (𝑋,𝜇,𝑇_𝑀), where 𝜇_𝑥(𝑡)=‖𝑥‖,‖𝑥‖,0𝑡≤1𝑡> (4.13) is a non-Archimedean RN-space.

Example 4.10. Let (𝑋,‖⋅‖) be a non-Archimedean normed linear space. Define 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖,∀𝑥∈𝑋,𝑡>0. (4.14) Then (𝑋,𝜇,𝑇_𝑀) is a non-Archimedean RN-space.

Definition 4.11. Let (𝑋,𝜇,𝑇) be a non-Archimedean RN-space. Let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} is said to be convergent if there exists 𝑥∈𝑋 such that lim_𝑛→∞𝜇_{𝑥𝑛−𝑥}(𝑡)=1 (4.15) for all 𝑡>0. In that case, 𝑥 is called the limit of the sequence {𝑥_𝑛}.
A sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if for each 𝜀>0 and 𝑡>0 there exists 𝑛₀ such that for all 𝑛≥𝑛₀ and 𝑝>0 we have 𝜇_{𝑥𝑛+𝑝−𝑥𝑛}(𝑡)>1−𝜀.
If each Cauchy sequence is convergent, then the random normed space is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space.

Remark 4.12 (see [53]). Let (𝑋,𝜇,𝑇_𝑀) be a non-Archimedean RN-space, then 𝜇_{𝑥𝑛+𝑝−𝑥𝑛}(𝑡)≥min𝜇_{𝑥𝑛+𝑗+1−𝑥𝑛+𝑗}(𝑡)∶𝑗=0,1,2,…,𝑝−1. (4.16) So, the sequence {𝑥_𝑛} is a Cauchy sequence if for each 𝜀>0 and 𝑡>0 there exists 𝑛₀ such that for all 𝑛≥𝑛₀ we have 𝜇_{𝑥𝑛+1−𝑥𝑛}(𝑡)>1−𝜀. (4.17)

5. Topological Structure of Random Normed Spaces

Definition 5.1. Let (𝑋,𝜇,𝑇) be an RN-space. We define the open ball 𝐵_𝑥(𝑟,𝑡) and the closed ball 𝐵_𝑥[𝑟,𝑡] with center 𝑥∈𝑋 and radius 0<𝑟<1, 𝑡>0 as follows: 𝐵_𝑥(𝑟,𝑡)=𝑦∈𝑋∶𝜇_𝑥−𝑦(𝑡),𝐵>1−𝑟_𝑥[𝑟,𝑡]=𝑦∈𝑋∶𝜇_𝑥−𝑦(𝑡)≥1−𝑟. (5.1)

Theorem 5.2. Let (𝑋,𝜇,𝑇) be an RN-space. Every open ball 𝐵_𝑥(𝑟,𝑡) is an open set.

Proof. Let 𝐵_𝑥(𝑟,𝑡)be an open ball with center 𝑥 and radius 𝑟 with respect 𝑡. Let 𝑦∈𝐵_𝑥(𝑟,𝑡). Then 𝜇_𝑥−𝑦(𝑡)>1−𝑟. Since 𝜇_𝑥−𝑦(𝑡)>1−𝑟, there exists 𝑡₀∈(0,𝑡) such that 𝜇_x−𝑦(𝑡₀)>1−𝑟. Put 𝑟₀=𝜇_𝑥,𝑦(𝑡₀). Since 𝑟₀>1−𝑟, there exists 𝑠∈(0,1) such that 𝑟₀>1−𝑠>1−𝑟. Now for given 𝑟₀ and 𝑠 such that 𝑟₀>1−𝑠, there exists 𝑟₁∈(0,1) such that 𝑇(𝑟₀,𝑟₁)>1−𝑠. Consider the open ball 𝐵_𝑦(1−𝑟₁,𝑡−𝑡₀). We claim 𝐵_𝑦(1−𝑟₁,𝑡−𝑡₀)⊂𝐵_𝑥(𝑟,𝑡). Now, let 𝑧∈𝐵_𝑦(1−𝑟₁,𝑡−𝑡₀). Then 𝜇_𝑦−𝑧(𝑡−𝑡₀)>𝑟₁. Therefore, we have 𝜇_𝑥−𝑧(𝑡)≥𝑇𝜇_𝑥−𝑦𝑡₀,𝜇_𝑦−𝑧𝑡−𝑡₀≥𝑇𝑟₀,𝑟₁≥1−𝑠>1−𝑟. (5.2) Thus 𝑧∈𝐵_𝑥(𝑟,𝑡) and hence 𝐵_𝑦(1−𝑟₁,𝑡−𝑡₀)⊂𝐵_𝑥(𝑟,𝑡).

Different kinds of topologies can be introduced in a random normed space [4]. The (𝑟,𝑡)-topology is introduced by a family of neighborhoods 𝐵_𝑥(𝑟,𝑡)_{𝑥∈𝑋,𝑡>0,𝑟∈(0,1)}. (5.3) In fact, every random norm 𝜇 on 𝑋 generates a topology ((𝑟,𝑡)-topology) on 𝑋 which has as a base the family of open sets of the form 𝐵_𝑥(𝑟,𝑡)_{𝑥∈𝑋,𝑡>0,𝑟∈(0,1)}. (5.4)

Remark 5.3. Since {𝐵_𝑥(1/𝑛,1/𝑛)∶𝑛=1,2,3,…} is a local base at 𝑥, the (𝑟,𝑡)-topology is first countable.

Theorem 5.4. Every RN-space (𝑋,𝜇,𝑇) is a Hausdorff space.

Proof. Let (𝑋,𝜇,𝑇) be an RN-space. Let 𝑥 and 𝑦 be two distinct points in 𝑋 and 𝑡>0. Then 0<𝜇_𝑥−𝑦(𝑡)<1. Put 𝑟=𝜇_𝑥−𝑦(𝑡). For each 𝑟₀∈(𝑟,1), there exists 𝑟₁ such that 𝑇(𝑟₁,𝑟₁)≥𝑟₀. Consider the open balls 𝐵_𝑥(1−𝑟₁,𝑡/2) and 𝐵_𝑦(1−𝑟₁,𝑡/2). Then clearly 𝐵_𝑥(1−𝑟₁,𝑡/2)∩𝐵_𝑦(1−𝑟₁,𝑡/2)=∅. For, if there exists 𝑧∈𝐵_𝑥1−𝑟₁,𝑡2∩𝐵_𝑦1−𝑟₁,𝑡2, (5.5) then we have 𝑟=𝜇_𝑥−𝑦(𝑡)≥𝑇𝜇_𝑥−𝑧𝑡2,𝜇_𝑦−𝑧𝑡2≥𝑇𝑟₁,𝑟₁≥𝑟₀>𝑟, (5.6) which is a contradiction. Hence (𝑋,𝜇,𝑇) is a Hausdorff space.

Definition 5.5. Let (𝑋,𝜇,𝑇) be an RN-space. A subset 𝐴 of 𝑋 is said to be R-bounded if there exists 𝑡>0 and 𝑟∈(0,1) such that 𝜇_𝑥−𝑦(𝑡)>1−𝑟 for all 𝑥,𝑦∈𝐴.

Theorem 5.6. Every compact subset 𝐴 of an RN-space (𝑋,𝜇,𝑇) is R-bounded.

Proof. Let 𝐴 be a compact subset of an RN-space (𝑋,𝜇,𝑇). Fix 𝑡>0 and 0<𝑟<1. Consider an open cover {𝐵_𝑥(𝑟,𝑡)∶𝑥∈𝐴}. Since 𝐴 is compact, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴 such that 𝐴⊆^𝑛_𝑖=1𝐵_𝑥𝑖(𝑟,𝑡). (5.7) Let 𝑥,𝑦∈𝐴. Then 𝑥∈𝐵_𝑥𝑖(𝑟,𝑡) and 𝑦∈𝐵_𝑥𝑗(𝑟,𝑡) for some 𝑖,𝑗. Thus we have 𝜇_{𝑥−𝑥𝑖}(𝑡)>1−𝑟 and 𝜇_{𝑦−𝑥𝑗}(𝑡)>1−𝑟. Now, let 𝛼=min𝜇_{𝑥𝑖,𝑥𝑗}(𝑡)∶1≤𝑖,𝑗≤𝑛. (5.8) Then 𝛼>0. Now, we have 𝜇_𝑥−𝑦(3𝑡)≥𝑇²𝜇_{𝑥−𝑥𝑖}(𝑡),𝜇_{𝑥𝑖,𝑥𝑗}(𝑡),𝜇_{𝑦−𝑥𝑗}(𝑡)≥𝑇²(1−𝑟,1−𝑟,𝛼)>1−𝑠. (5.9) Taking 𝑡′=3𝑡, we have 𝜇_𝑥−𝑦(𝑡′)>1−𝑠 for all 𝑥,𝑦∈𝐴. Hence 𝐴 is R-bounded.

Remark 5.7. In an RN-space (𝑋,𝜇,𝑇) every compact set is closed and R-bounded.

Theorem 5.8 (see [4]). If (𝑋,𝜇,𝑇) is an RN-space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞𝜇_𝑥𝑛(𝑡)=𝜇_𝑥(𝑡) almost everywhere.

Theorem 5.9. Let (𝑋,𝜇,𝑇) be an RN-space such that every Cauchy sequence in 𝑋 has a convergent subsequence. Then (𝑋,𝜇,𝑇) is complete.

Proof. Let {𝑥_𝑛} be a Cauchy sequence and let {𝑥_𝑖𝑛} be a subsequence of {𝑥_𝑛} that converges to 𝑥. We prove that 𝑥_𝑛→𝑥. Let 𝑡>0 and 𝜖∈(0,1) such that 𝑇(1−𝑟,1−𝑟)≥1−𝜖. (5.10) Since {𝑥_𝑛} is a Cauchy sequence, there is 𝑛₀∈ℕ such that 𝜇_{𝑥𝑚−𝑥𝑛}(𝑡)>1−𝑟 (5.11) for all 𝑚,𝑛≥𝑛₀. Since 𝑥_𝑖𝑛→𝑥, there is positive integer 𝑖_𝑝 such that 𝑖_𝑝>𝑛₀ and 𝜇_{𝑥𝑖𝑝−𝑥}𝑡2>1−𝑟. (5.12) Then, if 𝑛≥𝑛₀, then we have 𝜇_{𝑥𝑛−𝑥}(𝑡)≥𝑇𝜇_{𝑥𝑛−𝑥𝑖𝑝}𝑡2,𝜇_{𝑥𝑖𝑝−𝑥}𝑡2>𝑇(1−𝑟,1−𝑟)≥1−𝜖. (5.13) Therefore, 𝑥_𝑛→𝑥 and hence (𝑋,𝜇,𝑇) is complete.

Lemma 5.10. Let (𝑋,𝜇,𝑇) be an RN-space. If one defines 𝐹_𝑥,𝑦(𝑡)=𝜇_𝑥−𝑦(𝑡), (5.14) then 𝐹 is a random (probabilistic) metric on 𝑋, which is called the random (probabilistic) metric induced by the random norm 𝜇.

Lemma 5.11. A random (probabilistic) metric 𝐹 which is induced by a random norm on a RN-space (𝑋,𝜇,𝑇) has the following properties for all 𝑥,𝑦,z∈𝑋 and every scalar 𝛼≠0:
(i)𝐹_{𝑥+𝑧,𝑦+𝑧}(𝑡)=𝐹_𝑥,𝑦(𝑡),
(ii)𝐹_{𝛼𝑥,𝛼𝑦}(𝑡)=𝐹_𝑥,𝑦(𝑡/|𝛼|).

Proof. We have 𝐹_{𝑥+𝑧,𝑦+𝑧}(𝑡)=𝜇_{(𝑥+𝑧)−(𝑦+𝑧)}(𝑡)=𝜇_𝑥−𝑦(𝑡)=𝐹_𝑥,𝑦(𝑡). (5.15) Also, we have 𝐹_{𝛼𝑥,𝛼𝑦}(𝑡)=𝜇_{𝛼𝑥−𝛼𝑦}(𝑡)=𝜇_𝑥−𝑦𝑡|𝛼|=𝐹_𝑥,𝑦𝑡|𝛼|. (5.16)

Lemma 5.12. If (𝑋,𝜇,𝑇) is an RN-space, then
(i) the function (𝑥,𝑦)→𝑥+𝑦 is continuous,
(ii) the function (𝛼,𝑥)→𝛼𝑥 is continuous.

Proof. If 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦, then as 𝑛→∞𝜇_{(𝑥𝑛+𝑦𝑛)−(𝑥+𝑦)}(𝑡)≥𝑇𝜇_{𝑥𝑛−𝑥}𝑡2,𝜇_{𝑦𝑛−𝑦}𝑡2⟶1. (5.17) This proves (i).
Now, if 𝑥_𝑛→𝑥, 𝛼_𝑛→𝛼 and 𝛼_𝑛≠0, then 𝜇_{𝛼𝑛𝑥𝑛−𝛼𝑥}(𝑡)=𝜇_{𝛼𝑛(𝑥𝑛−𝑥)+𝑥(𝛼𝑛−𝛼)}(𝑡)≥𝑇𝜇_{𝛼𝑛(𝑥𝑛−𝑥)}𝑡2𝜇_{𝑥(𝛼𝑛−𝛼)}𝑡2=𝑇𝜇_{𝑥𝑛−𝑥}𝑡2𝛼_𝑛,𝜇_𝑥𝑡2𝛼_𝑛−𝛼⟶1 (5.18) as 𝑛→∞ and this proves (ii).

Definition 5.13. The RN-space (𝑋,𝜇,𝑇) is said to be a random Banach space whenever 𝑋 is complete with respect to the random metric induced by random norm.

Lemma 5.14. Let (𝑋,𝜇,𝑇) be an RN-space and define 𝐸_𝜆,𝜇∶𝑋→ℝ⁺∪{0} (5.19) by 𝐸_𝜆,𝜇(𝑥)=inf𝑡>0∶𝜇_𝑥(𝑡)>1−𝜆 (5.20) for each 𝜆∈(0,1) and 𝑥∈𝑋. Then we have
(i)𝐸_𝜆,𝜇(𝛼𝑥)=|𝛼|𝐸_𝜆,𝜇(𝑥) for every 𝑥∈𝑋 and 𝛼∈ℝ;
(ii) if 𝑇 satisfies (2.15), then for any 𝛾∈(0,1), there is 𝜆∈(0,1) such that 𝐸_𝛾,𝜇𝑥₁+⋯+𝑥_𝑛≤𝐸_𝜆,𝜇𝑥₁+⋯+𝐸_𝜆,𝜇𝑥_𝑛 (5.21) for any 𝑥,𝑦∈𝑋;
(iii) the sequence {𝑥_𝑛}_𝑛∈ℕ is convergent with respect to a random norm 𝜇 if and only if 𝐸_𝜆,𝜇(𝑥_𝑛−𝑥)→0. Also the sequence {𝑥_𝑛} is a Cauchy sequence with respect to a random norm 𝜇 if and only if it is a Cauchy sequence with 𝐸_𝜆,𝜇.

Proof. For (i), we find 𝐸_𝜆,𝜇(𝛼𝑥)=inf𝑡>0∶𝜇_𝛼𝑥(𝑡)>1−𝜆=inf𝑡>0∶𝜇_𝑥𝑡|𝛼|=|𝛼|>1−𝜆inf𝑡>0∶𝜇_𝑥(𝑡)=|𝛼|𝐸>1−𝜆_𝜆,𝜇(𝑥). (5.22) For (ii), by (2.15), for every 𝛼∈(0,1) we can find 𝜆∈(0,1) such that 𝑇^𝑛−1(1−𝜆,…,1−𝜆)≥1−𝛼. (5.23) Thus we have 𝜇_{𝑥1+⋯+𝑥𝑛}𝐸_𝜆,𝜇𝑥₁+⋯+𝐸_𝜆,𝜇𝑥_𝑛+𝑛𝛿≥_𝐿𝑇^𝑛−1𝜇_𝑥1𝐸_𝜆,ℳ𝑥₁+𝛿,…,𝜇_𝑥𝑛𝐸_𝜆,𝒫𝑥_𝑛()+𝛿≥𝑇1−𝜆,…,1−𝜆≥1−𝛼 (5.24) for every 𝛿>0, which implies that 𝐸_𝛼,𝜇𝑥₁+⋯+𝑥_𝑛≤𝐸_𝜆,𝜇𝑥₁+⋯+𝐸_𝜆,𝜇𝑥_𝑛+𝑛𝛿. (5.25) Since 𝛿>0 is arbitrary, we have 𝐸_𝛼,𝜇𝑥₁+⋯+𝑥_𝑛≤𝐸_𝜆,𝜇𝑥₁+⋯+𝐸_𝜆,𝜇𝑥_𝑛. (5.26)
For (iii), note that since 𝜇 is continuous, 𝐸_𝜆,𝜇(𝑥) is not an element of the set {𝑡>0∶𝜇_𝑥(𝑡)>1−𝜆} as soon as 𝑥≠0. Hence we have 𝜇_{𝑥𝑛−𝑥}𝜂>1−𝜆⟺𝐸_𝜆,𝜇𝑥_𝑛−𝑥<𝜂 (5.27) for every 𝜂>0. This completes the proof.

Definition 5.15. A function 𝑓 from an RN-space (𝑋,𝜇,𝑇) to an RN-space (𝑌,𝜈,𝑇′) is said to be uniformly continuous if for given 𝑟∈(0,1) and 𝑡>0, there exist 𝑟₀∈(0,1) and 𝑡₀>0 such that 𝜇_𝑥−𝑦𝑡₀>1−𝑟₀ (5.28) implies 𝜈_{𝑓(𝑥),𝑓(𝑦)}(𝑡)>1−𝑟. (5.29)

Theorem 5.16 (uniform continuity theorem). If 𝑓 is continuous function from a compact RN-space (𝑋,𝜇,𝑇) to an RN-space (𝑌,𝜈,𝑇′), then 𝑓 is uniformly continuous.

Proof. Let 𝑠∈(0,1) and 𝑡>0 be given. Then we can find 𝑟∈(0,1) such that 𝑇^′(1−𝑟,1−𝑟)>1−𝑠. (5.30) Since 𝑓∶𝑋→𝑌 is continuous, for each 𝑥∈𝑋, we can find 𝑟_𝑥∈(0,1) and 𝑡_𝑥>0 such that 𝜇_𝑥−𝑦𝑡_𝑥>1−𝑟_𝑥 (5.31) implies 𝜈_{𝑓(𝑥)−𝑓(𝑦)}𝑡2>1−𝑟. (5.32) But 𝑟_𝑥∈(0,1) and then we can find 𝑠_𝑥<𝑟_𝑥 such that 𝑇1−𝑠_𝑥,1−𝑠_𝑥>1−𝑟_𝑥. (5.33) Since 𝑋 is compact and 𝐵_𝑥𝑠_𝑥,𝑡_𝑥2∶𝑥∈𝑋 (5.34) is an open covering of 𝑋, there exist 𝑥₁,𝑥₂,…,𝑥_𝑘 in 𝑋 such that 𝑋=^𝑘_𝑖=1𝐵_𝑥𝑖𝑠_𝑥𝑖,𝑡_𝑥𝑖2. (5.35) Put 𝑠₀=min𝑠_𝑥𝑖 and 𝑡₀=min(𝑡_𝑥𝑖/2), 𝑖=1,2,…,𝑘. For any 𝑥,𝑦∈𝑋, if 𝜇_𝑥−𝑦𝑡₀>1−𝑠₀, (5.36) then 𝜇_𝑥−𝑦𝑡_𝑥𝑖2>1−𝑠_𝑥𝑖. (5.37) Since 𝑥∈𝑋, there exists 𝑥_𝑖 such that 𝜇_{𝑥−𝑥𝑖}𝑡_𝑥𝑖2>1−𝑠_𝑥𝑖. (5.38) Hence we have 𝜈_{𝑓(𝑥),𝑓(𝑥𝑖)}𝑡2>1−𝑟. (5.39) Now, since we have 𝜇_{𝑦−𝑥𝑖}𝑡_𝑥𝑖≥𝑇𝜇_𝑥−𝑦𝑡_𝑥𝑖2,𝜇_{𝑥−𝑥𝑖}𝑡_𝑥𝑖2≥𝑇1−𝑠_𝑥𝑖,1−𝑠_𝑥𝑖>1−𝑟_𝑥𝑖, (5.40) it follows that 𝜈_{𝑓(𝑦)−𝑓(𝑥𝑖)}𝑡2>1−𝑟. (5.41) Now, we have 𝜈_{𝑓(𝑥)−𝑓(𝑦)}(𝑡)≥𝑇𝜈_{𝑓(𝑥)−𝑓(𝑥𝑖)}𝑡2,𝜈_{𝑓(𝑦)−𝑓(𝑥𝑖)}𝑡2≥𝑇(1−𝑟,1−𝑟)>1−𝑠. (5.42) Hence 𝑓 is uniformly continuous.

Remark 5.17. Let 𝑓 be a uniformly continuous function from RN-space (𝑋,𝜇,𝑇) to RN-space (𝑌,𝜈,𝑇^′). If {𝑥_𝑛} is a Cauchy sequence in 𝑋, then {𝑓(𝑥_𝑛)} is also a Cauchy sequence in 𝑌.

Theorem 5.18. Every compact RN-space is separable.

Proof. Let (𝑋,𝜇,𝑇) be the given compact RN-space. Let 𝑟∈(0,1) and 𝑡>0. Since 𝑋 is compact, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛 in 𝑋 such that 𝑋=^𝑛_𝑖=1𝐵_𝑥𝑖(𝑟,𝑡). (5.43) In particular, for each 𝑛∈ℕ, we can choose a finite subset 𝐴_𝑛 such that 𝑋=_{𝑎∈𝐴𝑛}𝐵_𝑎𝑟_𝑛,1𝑛, (5.44) in which 𝑟_𝑛∈(0,1). Let 𝐴=_𝑛∈ℕ𝐴_𝑛. (5.45) Then 𝐴 is countable. We claim that 𝑋⊂𝐴. Let 𝑥∈𝑋. Then for each 𝑛∈ℕ, there exists 𝑎_𝑛∈𝐴_𝑛 such that 𝑥∈𝐵_𝑎𝑛(𝑟_n,1/𝑛). Thus 𝑎_𝑛 converges to 𝑥. But since 𝑎_𝑛∈𝐴 for all 𝑛, 𝑥∈𝐴. Hence 𝐴 is dense in 𝑋 and thus 𝑋 is separable.

Definition 5.19. Let 𝑋 be any nonempty set and (𝑌,𝜈,𝑇′) be an RN-space. Then a sequence {𝑓_n} of functions from 𝑋 to 𝑌 is said to be uniformly convergent to a function 𝑓 from 𝑋 to 𝑌 if for given 𝑟∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜈_{𝑓𝑛(𝑥)−𝑓(𝑥)}(𝑡)>1−𝑟 (5.46) for all 𝑛≥𝑛₀ and 𝑥∈𝑋.

Definition 5.20. A family ℱ of functions from an RN-space (𝑋,𝜇,𝑇) to a complete RN-space (𝑌,𝜈,𝑇′) is said to be equicontinuous if for any 𝑟∈(0,1) and 𝑡>0, there exist 𝑟₀∈(0,1) and 𝑡₀>0 such that 𝜇_𝑥−𝑦𝑡₀>1−𝑟₀⟹𝜈_{𝑓(𝑥)−𝑓(𝑦)}(t)>1−𝑟 (5.47) for all 𝑓∈ℱ.

Lemma 5.21. Let {𝑓_𝑛} be an equicontinuous sequence of functions from an RN-space (𝑋,𝜇,𝑇) to a complete RN-space (𝑌,𝜈,𝑇′). If {𝑓_𝑛} converges for each point of a dense subset 𝐷 of 𝑋, then {𝑓_𝑛} converges for each point of 𝑋 and the limit function is continuous.

Proof. Let 𝑠∈(0,1) and 𝑡>0 be given. Then we can find 𝑟∈(0,1) such that 𝑇^′2(1−𝑟,1−𝑟,1−𝑟)>1−𝑠. (5.48) Since ℱ={𝑓_𝑛} is equicontinuous family, for given 𝑟∈(0,1) and 𝑡>0, there exist 𝑟₁∈(0,1) and 𝑡₁>1 such that, for each 𝑥,𝑦∈𝑋, 𝜇_𝑥−𝑦𝑡₁>1−𝑟₁⟹𝜈_{𝑓𝑛(𝑥)−𝑓𝑛(𝑦)}𝑡3>1−𝑟 (5.49) for all 𝑓_𝑛∈ℱ. Since 𝐷 is dense in 𝑋, there exists 𝑦∈𝐵_𝑥𝑟₁,𝑡₁∩𝐷, (5.50) and {𝑓_𝑛(𝑦)} converges for that 𝑦. Since {𝑓_𝑛(𝑦)} is a Cauchy sequence, for any 𝑟∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜈_{𝑓𝑛(𝑦)−𝑓𝑚(𝑦)}𝑡3>1−𝑟 (5.51) for all 𝑚,𝑛≥𝑛₀. Now, for any 𝑥∈𝑋, we have 𝜈_{𝑓𝑛(𝑥)−𝑓𝑚(𝑥)}(𝑡)≥𝑇′²𝜈_{𝑓𝑛(𝑥)−𝑓𝑛(𝑦)}𝑡3,𝜈_{𝑓𝑛(𝑦)−𝑓𝑚(𝑦)}𝑡3,𝜈_{𝑓𝑚(𝑥)−𝑓𝑚(𝑦)}𝑡3≥𝑇′²(1−𝑟,1−𝑟,1−𝑟)>1−𝑠. (5.52) Hence {𝑓_𝑛(𝑥)} is a Cauchy sequence in 𝑌. Since 𝑌 is complete, 𝑓_𝑛(𝑥) converges. Let 𝑓(𝑥)=lim𝑓_𝑛(𝑥). We claim that 𝑓 is continuous. Let 𝑠_𝑜∈1−𝑟 and 𝑡₀>0 be given. Then we can find 𝑟₀∈1−𝑟 such that 𝑇^′21−𝑟₀,1−𝑟₀,1−𝑟₀>1−𝑠₀. (5.53) Since ℱ is equicontinuous, for given 𝑟₀∈(0,1) and 𝑡₀>0, there exist 𝑟₂∈(0,1) and 𝑡₂>0 such that 𝜇_𝑥−𝑦𝑡₂>1−𝑟₂⟹𝜈_{𝑓𝑛(𝑥)−𝑓𝑛(𝑦)}𝑡₀3>1−𝑟₀ (5.54) for all 𝑓_𝑛∈ℱ. Since 𝑓_𝑛(𝑥) converges to 𝑓(𝑥), for given 𝑟₀∈(0,1) and 𝑡₀>0, there exists 𝑛₁∈ℕ such that 𝜈_{𝑓𝑛(𝑥)−𝑓(𝑥)}𝑡₀3>1−𝑟₀. (5.55) Also since 𝑓_𝑛(𝑦) converges to 𝑓(𝑦), for given 𝑟₀∈(0,1) and 𝑡₀>0, there exists 𝑛₂∈ℕ such that 𝜈_{𝑓𝑛(𝑦)−𝑓(𝑦)}𝑡₀3>1−𝑟₀ (5.56) for all 𝑛≥𝑛₂. Now, for all 𝑛≥max{𝑛₁,𝑛₂}, we have 𝜈_{𝑓(𝑥)−𝑓(𝑦)}𝑡₀≥𝑇^′2𝜈_{𝑓(𝑥)−𝑓𝑛(𝑥)}𝑡₀3,𝜈_{𝑓𝑛(𝑥)−𝑓𝑛(𝑦)}𝑡₀3,𝜈_{𝑓𝑛(𝑦)−𝑓(𝑦)}𝑡₀3≥𝑇^′21−𝑟₀,1−𝑟₀,1−𝑟₀>1−𝑠₀. (5.57) Hence 𝑓 is continuous.

Theorem 5.22 (Ascoli-Arzela Theorem). Let (𝑋,𝜇,𝑇) be a compact RN-space and (𝑌,𝜈,𝑇′) be a complete RN-space. Let ℱ be an equicontinuous family of functions from 𝑋 to 𝑌. If {𝑓_𝑛}_𝑛∈ℕ is a sequence in ℱ such that {𝑓_𝑛(𝑥)∶𝑛∈ℕ} (5.58) is a compact subset of 𝑌 for each 𝑥∈𝑋, then there exists a continuous function 𝑓 from 𝑋 to 𝑌 and a subsequence {𝑔_𝑛} of {𝑓_𝑛} such that 𝑔_𝑛 converges uniformly to 𝑓 on 𝑋.

Proof. Since (𝑋,𝜇,𝑇) be a compact RN-space, by Theorem 5.18, 𝑋 is separable. Let 𝐷=𝑥_𝑖∶𝑖=1,2,… (5.59) be a countable dense subset of 𝑋. By hypothesis, for each 𝑖, {𝑓_𝑛(𝑥_𝑖)∶𝑛∈ℕ} (5.60) is compact subset of 𝑌. Since every RN-space is first countable space, every compact subset of 𝑌 is sequentially compact. Thus by standard argument, we have a subsequence {𝑔_𝑛} of {𝑓_𝑛} such that {𝑔_𝑛(𝑥_𝑖)} converges for each 𝑖=1,2,…. By Lemma 5.21, there exists a continuous function 𝑓 from 𝑋 to 𝑌 such that {𝑔_𝑛(𝑥)} converges to 𝑓(𝑥) for all 𝑥∈𝑋.
Now we claim that {𝑔_𝑛} converges uniformly to 𝑓 on 𝑋. Let 𝑠∈(0,1) and 𝑡>0 be given. Then we can find 𝑟∈(0,1) such that 𝑇′²(1−𝑟,1−𝑟,1−𝑟)>1−𝑠. (5.61) Since ℱ is equicontinuous, there exist 𝑟₁∈(0,1) and 𝑡₁>0 such that 𝜇_𝑥−𝑦𝑡₁>1−𝑟₁⟹𝜈_{𝑔𝑛(𝑥),𝑔𝑛(𝑦)}𝑡3>1−𝑟 (5.62) for all 𝑛∈ℕ. Since 𝑋 is compact, by Theorem 5.16, 𝑓 is uniformly continuous. Hence for given 𝑟∈(0,1) and 𝑡>0, there exist 𝑟₂∈(0,1) and 𝑡₂>0 such that 𝜇_𝑥−𝑦𝑡₂>1−𝑟₂⟹𝜈_{𝑓(𝑥)−𝑓(𝑦)}𝑡3>1−𝑟 (5.63) for all 𝑥,𝑦∈𝑋. Let 𝑟₀=min{𝑟₁,𝑟₂} and 𝑡₀=min{𝑡₁,𝑡₂}. Since 𝑋 is compact and 𝐷 is dense in 𝑋, 𝑋=^𝑘_𝑖=1𝐵_𝑥𝑖𝑟₀,𝑡₀ (5.64) for some 𝑘. Thus for each 𝑥∈𝑋, there exists 𝑖, 𝑖≤𝑖≤𝑘, such that 𝜇_{𝑥−𝑥𝑖}𝑡₀>1−𝑟₀. (5.65) But since 𝑟₀=min{𝑟₁,𝑟₂} and 𝑡₀=min{𝑡₁,𝑡₂}, we have, by the equicontinuity of ℱ, 𝜈_{𝑔𝑛(𝑥)−𝑔𝑛(𝑥𝑖)}𝑡3>1−𝑟 (5.66) and we also have, by the uniform continuity of 𝑓, 𝜈_{𝑓(𝑥)−𝑓(𝑥𝑖)}𝑡3>1−𝑟. (5.67) Since {𝑔_𝑛(𝑥_𝑗)} converges to 𝑓(𝑥_𝑗), for any 𝑟∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜈_{𝑔𝑛(𝑥𝑗)−𝑓(𝑥𝑗)}𝑡3>1−𝑟 (5.68) for all 𝑗=1,2,…,𝑛.
Now, for each 𝑥∈𝑋, we have 𝜈_{𝑔𝑛(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇^′2𝜈_{𝑔𝑛(𝑥)−𝑔𝑛(𝑥𝑖)}𝑡3,𝜈_{𝑔𝑛(𝑥𝑖)−𝑓(𝑥𝑖)}𝑡3,𝜈_{𝑓(𝑥𝑖)−𝑓(𝑥)}𝑡3≥𝑇^′2(1−𝑟,1−𝑟,1−𝑟)>1−𝑠. (5.69) Hence {𝑔_𝑛} converges uniformly to 𝑓 on 𝑋.

Lemma 5.23. A subset 𝐴 of ℝ is R-bounded in (ℝ,𝜇,𝑇) if and only if it is bounded in ℝ.

Proof. Let 𝐴 be a subset in ℝ which is R-bounded in (ℝ,𝜇,𝑇), then there exist 𝑡₀>0 and 𝑟₀∈(0,1) such that 𝜇_𝑥(𝑡₀)>1−𝑟₀ for each 𝑥∈𝐴 and therefore we have 𝑡₀≥𝐸_𝑟0,𝜇(𝑥)=|𝑥|𝐸_𝑟0,𝜇(1). (5.70) Now, 𝐸_𝑟0,𝜇(1)≠0. If we put 𝑘=𝑡₀/(𝐸_𝑟0,𝜇(1)), then we have |𝑥|≤𝑘 for each 𝑥∈𝐴, that is, 𝐴 is bounded in ℝ. The converse is easy to see.

Lemma 5.24. A sequence {𝛽_𝑛}_𝑛∈ℕ is convergent in the RN-space (ℝ,𝜇,𝑇) if and only if it is convergent in (ℝ,|⋅|).

Proof. Let 𝛽_𝑛→𝛽 in ℝ, by Lemma 5.14 (i), we have 𝐸_𝜆,𝜇𝛽_𝑛−𝛽=𝛽_𝑛−𝛽𝐸_𝜆,𝜇(1)⟶0. (5.71) Then, by Lemma 5.14 (iii), 𝛽→−_𝑛^𝜇𝛽.
Conversely, let 𝛽→−_𝑛^𝜇𝛽, then, by Lemma 5.14, we have lim_𝑛→+∞𝛽_𝑛−𝛽𝐸_𝜆,𝜇(1)=lim_𝑛→+∞𝐸_𝜆,𝜇𝛽_𝑛−𝛽=0. (5.72) Now, 𝐸_𝜆,𝜇(1)≠0 and so 𝛽_𝑛→𝛽 in ℝ.

Corollary 5.25. If the real sequence {𝛽_𝑛}_𝑛∈ℕ is R-bounded, then it has at least one limit point.

Definition 5.26. The 3-tuple (ℝ^𝑛,Φ,𝑇) is called a random Euclidean normed space if 𝑇 is a continuous 𝑡-norm and Φ_𝑥(𝑡) is a random Euclidean norm defined by Φ_𝑥(𝑡)=^𝑛_𝑗=1𝜇_𝑥𝑗(𝑡), (5.73) where ∏^𝑛_𝑗=1𝑎_𝑗=𝑇′^𝑛−1(𝑎₁,…,𝑎_𝑛), 𝑇′≫𝑇, 𝑥=(𝑥₁,…,𝑥_𝑛), 𝑡>0, and 𝜇 is a random norm.
For example, let Φ_𝑥(𝑡)=exp(‖𝑥‖/𝑡)⁻¹, 𝜇_𝑥𝑗(𝑡)=exp(|𝑥_𝑗|/𝑡)⁻¹, and 𝑇=min. Then we have Φ_𝑥(𝑡)=min_𝑗𝜇_𝑥𝑗(𝑡) or equivalently ‖𝑥‖=max_𝑗|𝑥_𝑗|.

Lemma 5.27. Suppose that the hypotheses of Definition 5.26 are satisfied. Then (ℝ^𝑛,Φ,𝑇) is an RN-space.

Proof . The properties of (RN1) and (RN2) follow immediately from the definition. For the triangle inequality (RN3), suppose that 𝑥,𝑦∈𝑋 and 𝑡,𝑠>0. Then 𝑇Φ_𝑥(𝑡),Φ_𝑦(𝑠)=𝑇^𝑛_𝑗=1𝒫_𝑥𝑗(𝑡),^𝑛_𝑗=1𝒫_𝑦𝑗(𝑠)=𝑇𝑇^′𝑛−1𝒫_𝑥1(𝑡),…,𝒫_𝑥𝑛(𝑡),𝑇^′𝑛−1𝒫_𝑦1(𝑡),…,𝒫_𝑦𝑛(𝑡)≤𝑇^′𝑛−1𝑇𝒫_𝑥1(𝑡),𝒫_𝑦1(𝑡),…,𝑇𝒫_𝑥𝑛(𝑡),𝒫_𝑦𝑛(𝑡)≤𝑇^′𝑛−1𝒫_𝑥1+𝑦1(𝑡+𝑠),…,𝒫_{𝑥𝑛+𝑦𝑛}()=𝑡+𝑠_𝑛𝑗=1𝒫_{𝑥𝑗+𝑦𝑗}(𝑡+𝑠)=Φ_𝑥+𝑦(𝑡+𝑠). (5.74)

Lemma 5.28. Suppose that (ℝ^𝑛,Φ,𝑇) is a random Euclidean normed space and 𝐴 is an infinite and R-bounded subset of ℝ^𝑛. Then 𝐴 has at least one limit point.

Proof. Let {𝑥^(𝑚)}_𝑚∈ℕ⊆𝐴 be an infinite sequence. Since 𝐴 is R-bounded and so is {𝑥^(𝑚)}_𝑚∈ℕ. Therefore there exist 𝑡₀>0 and 𝑟₀∈(0,1) such that 1−𝑟₀<Φ_𝑥(𝑡₀) for each 𝑥∈𝐴, which implies that 𝐸_𝑟0,Φ(𝑥)≤𝑡₀. However, we have 𝐸_𝑟0,Φ(𝑥)=inf𝑡>0∶1−𝑟₀<Φ_𝑥(𝑡)=inf𝑡>0∶1−𝑟₀<^𝑛_𝑗=1𝜇_𝑥𝑗(𝑡)≥inf𝑡>0∶1−𝑟₀<𝜇_𝑥𝑗(𝑡)=𝐸_𝑟0,𝜇𝑥_𝑗 (5.75) for each 1≤𝑗≤𝑛. Therefore |𝑥_𝑗|≤𝑘 in which 𝑘=𝑡₀/(𝐸_𝑟0,𝜇(1)), that is, the real sequences {𝑥_𝑗^(𝑚)} for all 𝑗∈{1,…,𝑛} are bounded. Hence there exists a subsequence {𝑥^(𝑚𝑘1)₁} which converges to 𝑥₁ in 𝐴 w.r.t. the random norm 𝜇. The corresponding sequence {𝑥^(𝑚𝑘1)₂} is bounded and so there exists a subsequence {𝑥^(𝑚𝑘2)₂} of {𝑥^(𝑚𝑘1)₂} which converges to 𝑥₂ with respect to the random norm 𝜇. Continuing like this, we find a subsequence {𝑥^(𝑚𝑘)} converging to 𝑥=(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛.

Lemma 5.29. Let (ℝ^𝑛,Φ,𝑇) be a random Euclidean normed space. Let {𝑄₁,𝑄₂,…} be a countable collection of nonempty subsets in ℝ^𝑛 such that 𝑄_𝑘+1⊆𝑄_𝑘, each 𝑄_𝑘 is closed and 𝑄₁ is R-bounded. Then ∩^∞_𝑘=1𝑄_𝑘 is nonempty and closed.

Proof. Using the above lemma, the proof proceeds as in the classical case [see Theorem 3.25 in [54]].

We call an 𝑛-dimensional ball 𝐵_𝑥(𝑟,𝑡) a rational ball if 𝑥∈ℚ^𝑛, 𝑟₀∈(0,1), and 𝑡∈ℚ⁺.

Theorem 5.30. Let (ℝ^𝑛,Φ,𝑇) be a random Euclidean normed space in which 𝑇 satisfies (2.15). Let 𝐺={𝐴₁,𝐴₂,…} be a countable collection of 𝑛-dimensional rational open balls. If 𝑥∈ℝ^𝑛 and 𝑆 is an open subset of ℝ^𝑛 containing 𝑥, then there exists a 𝐴_𝑘∈𝐺 such that 𝑥∈𝐴_𝑘⊆𝑆, for some 𝑘≥1.

Proof. Since 𝑥∈𝑆 and 𝑆 is an open, there exist 𝑟∈(0,1) and 𝑡>0 such that 𝐵_𝑥(𝑟,𝑡)⊆𝑆. By (2.15), we can find 𝜂∈(0,1) such that 1−𝑟<𝑇(1−𝜂,1−𝜂). Let {𝜉_𝑘}^𝑛_𝑘=1 be a finite sequence such that 1−𝜂<∏^𝑛_𝑘=1(1−𝜉_𝑘) and 𝑥=(𝑥₁,…,𝑥_𝑛) and then we can find 𝑦=(𝑦₁,…,𝑦_𝑛)∈ℚ^𝑛 such that 1−𝜉_𝑘)<𝜇_{𝑥𝑘−𝑦𝑘}(𝑡/2). Therefore, we have 1−𝜂<^𝑛_𝑘=11−𝜉_𝑘≤Φ_𝑥−𝑦𝑡2=^𝑛_𝑘=1𝜇_{𝑥𝑘−𝑦𝑘}𝑡2 (5.76) and so 𝑥∈𝐵_𝑦(𝜂,𝑡/2). Now we prove that 𝐵_𝑦(𝜂,𝑡/2)⊆𝐵_𝑥(𝑟,𝑡). Let 𝑧∈𝐵_𝑦(𝜂,𝑡/2). Then Φ_𝑦−𝑧(𝑡/2)>1−𝜂 and hence 1−𝑟<𝑇1−𝜂,1−𝜂≤𝑇Φ_𝑥−𝑦𝑡2,Φ_𝑦−𝑧𝑡2≤Φ_𝑥−𝑧(𝑡). (5.77) On the other hand, there exists 𝑡₀∈ℚ such that 𝑡₀<𝑡/2 and 𝑥∈𝐵_𝑦(𝜂,𝑡₀)⊆𝐵_𝑦(𝜂,𝑡/2)⊆𝐵_𝑥(𝑟,𝑡)⊆𝑆. Now 𝐵_𝑦(𝜂,𝑡₀)∈𝐺 and the proof is complete.

Corollary 5.31. In a random Euclidean normed space (ℝ^𝑛,Φ,𝑇) in which 𝑇 satisfies (2.15), every closed and R-bounded set is compact.

Proof. The proof is similar to the proof of Theorem 3.29 in [54].

Corollary 5.32. Let (ℝ^𝑛,Φ,𝑇) be a random Euclidean normed space in which 𝑇 satisfies (2.15) and 𝑆⊆ℝ^𝑛. Then 𝑆 is compact set if and only if it is R-bounded and closed.

Corollary 5.33. The random Euclidean normed space (ℝ^𝑛,Φ,𝑇) is complete.

Proof. Let {𝑥_𝑚} be a Cauchy sequence in the random Euclidean normed space (ℝ^𝑛,Φ,𝑇). Since 𝐸_𝜆,Φ𝑥_𝑛−𝑥_𝑚=inf𝑡>0∶Φ_{𝑥𝑛−𝑥𝑚}(𝑡)>1−𝜆=inf𝑡>0∶^𝑛_𝑗=1𝒫_{𝑥𝑚,𝑗−𝑥𝑛,𝑗}(𝑡)>1−𝜆≥inf𝑡>0∶𝒫_{𝑥𝑚,𝑗−𝑥𝑛,𝑗}(𝑡)>1−𝜆=𝐸_𝜆,𝒫𝑥_𝑚,𝑗−𝑥_𝑛,𝑗=𝑥_𝑚,𝑗−𝑥_𝑛,𝑗𝐸_𝜆,𝒫(1), (5.78) the sequence {𝑥_𝑚,𝑗} in which 𝑗=1,…,𝑛 is a Cauchy sequence in ℝ and convergent to 𝑥_𝑗∈ℝ then, and by Lemma 5.14, the sequence {𝑥_𝑚,𝑗} is convergent in RN-space (ℝ,𝜇,𝑇). We prove that {𝑥_𝑚} is convergent to 𝑥=(𝑥₁,…,𝑥_𝑛) and lim_𝑚→∞Φ_{𝑥𝑚−𝑥}(𝑡)=lim^𝑛_{𝑚→∞𝑗=1}𝒫_{𝑥𝑚,𝑗−𝑥𝑗}(𝑡)=𝑇^′𝑛−1(1,…,1)=1. (5.79)

6. Random Functional Analysis

Theorem 6.1. Let {𝑥₁,…,𝑥_𝑛} be a linearly independent set of vectors in vector space 𝑋 and (𝑋,𝜇,𝑇) be an RN-space. Then there is 𝑐≠0 and an RN-space (ℝ,𝜇^′,𝑇) such that for every choice of the 𝑛 real scalars 𝛼₁,…,𝛼_𝑛 we have 𝜇_{𝛼1𝑥1+⋯+𝛼𝑛𝑥𝑛}(𝑡)≤𝜇^′_𝑐^𝑛_𝑗=1𝛼_𝑗(𝑡). (6.1)

Proof. Put 𝑠=|𝛼₁|+⋯+|𝛼_𝑛|. If 𝑠=0, all 𝛼_𝑗s must be zero and so (6.1) holds for any 𝑐. Let 𝑠>0. Then (6.1) is equivalent to the inequality which we obtain from (6.1) by dividing by 𝑠 and putting 𝛽_𝑗=(𝛼_𝑗)/𝑠, that is, 𝜇_{𝛽1𝑥1+⋯+𝛽𝑛𝑥𝑛}𝑡^′≤𝜇^′_𝑐𝑡′, (6.2) where 𝑡𝑡′=𝑠 and ^𝑛∑_𝑗=1|𝛽_𝑗|=1. Hence it suffices to prove the existence of a 𝑐≠0 and random norm 𝜇^′ such that (6.2) holds. Suppose that this is not true. Then there exists a sequence {𝑦_𝑚} of vectors, 𝑦_𝑚=𝛽_1,𝑚𝑥₁+⋯+𝛽_𝑛,𝑚𝑥_𝑛, (6.3) where ^𝑛∑_𝑗=1|𝛽_𝑗,𝑚|=1, such that 𝜇_𝑦𝑚(𝑡)⟶1 (6.4) as 𝑚→∞ for every 𝑡>0. Since ∑^𝑛_𝑗=1|𝛽_𝑗,𝑚|=1, we have |𝛽_𝑗,𝑚|≤1 and then, by Lemma 5.23, the sequence of {𝛽_𝑗,𝑚} is R-bounded. According the Corollary 5.25, {𝛽_1,𝑚} has a convergent subsequence. Let 𝛽₁ denote the limit of that subsequence, and let {𝑦_1,𝑚} denote the corresponding subsequence of {𝑦_𝑚}. By the same argument, {𝑦_1,𝑚} has a subsequence {𝑦_2,𝑚} for which the corresponding of real scalars 𝛽₂^(𝑚) convergence; let 𝛽₂ denote the limit. Continuing this process, after 𝑛 steps, we obtain a subsequence {𝑦_𝑛,𝑚}_𝑚 of {𝑦_𝑚} such that 𝑦_𝑛,𝑚=^𝑛_𝑗=1𝛾_𝑗,𝑚𝑥_𝑗, (6.5) where ^𝑛∑_𝑗=1|𝛾_𝑗,𝑚|=1, and 𝛾_𝑗,𝑚→𝛽_𝑗 as 𝑚→∞. By the Lemma 5.14 (ii) for any 𝛼∈(0,1) there exists 𝜆∈(0,1) such that 𝐸_𝛼,𝜇𝑦_𝑛,𝑚−^𝑛_𝑗=1𝛽_𝑗𝑥_𝑗=𝐸_𝛼,𝜇^𝑛_𝑗=1𝛾_𝑗,𝑚−𝛽_𝑗𝑥_𝑗≤^𝑛_𝑗=1𝛾_𝑗,𝑚−𝛽_𝑗𝐸_𝜆,𝜇𝑥_𝑗⟶0 (6.6) as 𝑚→∞. By the Lemma 5.14 (iii), we conclude lim_𝑚→∞𝑦_𝑛,𝑚=^𝑛_𝑗=1𝛽_𝑗𝑥_𝑗, (6.7) where ^𝑛∑_𝑗=1|𝛽_𝑗|=1, so that not all 𝛽_𝑗 can be zero. Put 𝑦=∑^𝑛_𝑗=1𝛽_𝑗𝑥_𝑗. Since {𝑥₁,…,𝑥_𝑛} is a linearly independent set, we thus have 𝑦≠0. Since 𝜇_𝑦𝑚(𝑡)⟶1, (6.8) by assumption, we have 𝜇_{𝑦𝑛,𝑚}(𝑡)⟶1. (6.9) Hence it follows that 𝜇_𝑦(𝑡)=𝜇_{(𝑦−𝑦𝑛,𝑚)+𝑦𝑛,𝑚}(𝑡)≥𝑇𝜇_{𝑦−𝑦𝑛,𝑚}𝑡2,𝜇_{𝑦𝑛,𝑚}𝑡2⟶1 (6.10) and so 𝑦=0, which is a contradiction.

Definition 6.2. Let (𝑋,𝜇,𝑇) and (𝑋,𝜈,𝑇′) be two RN-spaces. Then two random norms 𝜇 and 𝜈 are said to be equivalent whenever 𝑥→−_𝑛^𝜇𝑥 in (𝑋,𝜇,𝑇) if and only if 𝑥→−_𝑛^𝜈𝑥 in (𝑋,𝜈,𝑇′).

Theorem 6.3. On a finite dimensional vector space 𝑋, every two random norms 𝜇 and 𝜈 are equivalent.

Proof. Let dim𝑋=𝑛 and {𝑣₁,…,𝑣_𝑛} be a basis for 𝑋. Then every 𝑥∈𝑋 has a unique representation 𝑥=∑^𝑛_𝑗=1𝛼_𝑗𝑣_𝑗. Let 𝑥→−_𝑚^𝜇𝑥 in (𝑋,𝜇,𝑇), but, for each 𝑚∈ℕ, 𝑥_𝑚 has a unique representation, that is, 𝑥_𝑚=𝛼_1,𝑚𝑣₁+⋯+𝛼_𝑛,𝑚𝑣_𝑛. (6.11) By Theorem 6.1, there is 𝑐≠0 and a random norm 𝜇^′ such that (6.1) holds. So 𝜇_{𝑥𝑚−𝑥}(𝑡)≤𝜇^′_{𝑐∑𝑛𝑗=1𝛼𝑗,𝑚−𝛼𝑗}(𝑡)≤𝜇^′_{𝑐𝛼𝑗,𝑚−𝛼𝑗}(𝑡). (6.12) Now, if 𝑚→∞, then 𝜇_{𝑥𝑚−𝑥}(𝑡)⟶1 (6.13) for every 𝑡>0 and hence |𝛼_𝑗,𝑚−𝛼_𝑗|→0 in ℝ. On the other hand, by Lemma 5.14 (ii), for any 𝛼∈(0,1), there exists 𝜆∈(0,1) such that 𝐸_𝛼,𝜈𝑥_𝑚−𝑥≤^𝑛_𝑗=1𝛼_𝑗,𝑚−𝛼_𝑗𝐸_𝜆,𝜈𝑣_𝑗. (6.14) Since |𝛼_𝑗,𝑚−𝛼_𝑗|→0, then we have 𝑥→−_𝑚^𝜈𝑥 in (𝑋,𝜈,𝑇′). With the same argument 𝑥_𝑚→𝑥 in (𝑋,𝜈,𝑇′) imply 𝑥_𝑚→𝑥 in (𝑋,𝜇,𝑇).

Definition 6.4. A linear operator Λ∶(𝑋,𝜇,𝑇)→(𝑌,𝜈,𝑇^′) is said to be random bounded if there exists a constant ℎ∈ℝ−{0} such that for every 𝑥∈𝑋 and 𝑡>0𝜈_Λ𝑥(𝑡)≥𝜇_ℎ𝑥(𝑡). (6.15)

Note that, by Lemma 5.14 and last definition, we have 𝐸_𝜆,𝜈(Λx)=inf𝑡>0∶𝜈_Λ𝑥(𝑡)>1−𝜆≤inf𝑡>0∶𝜇_𝑥𝑡|ℎ|>1−𝜆=|ℎ|inf𝑡>0∶𝜇_𝑥(𝑡)=|ℎ|𝐸>1−𝜆_𝜆,𝜇(𝑥). (6.16)

Theorem 6.5. Every linear operator Λ∶(𝑋,𝜇,𝑇)→(𝑌,𝜈,𝑇^′) is random bounded if and only if it is continuous.

Proof. By (6.15) every random bounded linear operator is continuous. Now, we prove that the converse it. Let the linear operator Λ be continuous but not random bounded. Then, for each 𝑛∈ℕ there is 𝑥_𝑛 in 𝑋 such that 𝐸_𝜆,𝜈(Λ𝑥_𝑛)≥𝑛𝐸_𝜆,𝜇(𝑝_𝑛). If we let 𝑦_𝑛=𝑥_𝑛𝑛𝐸_𝜆,𝜇𝑥_𝑛, (6.17) then it is easy to see 𝑦_𝑛→0 but Λ𝑦_𝑛 do not tend to 0.

Definition 6.6. A linear operator Λ∶(𝑋,𝜇,𝑇)→(𝑌,𝜈,𝑇^′) is called a random topological isomorphism if Λ is one-to-one and onto and both Λ and Λ⁻¹ are continuous. RN-spaces (𝑋,𝜇,𝑇) and (𝑌,𝜈,𝑇^′) for which such a Λ exists are said to be random topologically isomorphic.

Lemma 6.7. A linear operator Λ∶(𝑋,𝜇,𝑇)→(𝑌,𝜈,𝑇^′) is random topological isomorphism if Λ is onto and there exists constants 𝑎,𝑏≠0 such that 𝜇_𝑎𝑥(𝑡)≤𝜈_Λ𝑥(𝑡)≤𝜇_𝑏𝑥(𝑡).

Proof. By hypothesis Λ is random bounded and, by last theorem, is continuous and, since Λ𝑥=0, we have 1=𝜈_Λ𝑥(𝑡)≤𝜇_𝑥𝑡|𝑏| (6.18) and consequently 𝑥=0. So Λ is one-to-one. Thus Λ⁻¹ exists and, since 𝜈_Λ𝑥(𝑡)≤𝜇_𝑏𝑥(𝑡) (6.19) is equivalent to 𝜈_𝑦(𝑡)≤𝜇_{𝑏Λ−1𝑦}(𝑡)=𝜇_Λ−1𝑦𝑡|𝑏| (6.20) or 𝜈₍1𝑏^)𝑦(𝑡)≤𝜇_Λ−1𝑦(𝑡), (6.21) where 𝑦=Λ𝑥, we see that Λ⁻¹ is random bounded and, by last theorem, is continuous. Hence Λ is an random topological isomorphism.

Corollary 6.8. Random topologically isomorphism preserves completeness.

Theorem 6.9. Every linear operator Λ∶(𝑋,𝜇,𝑇)→(𝑌,𝜈,𝑇′) where dim𝑋<∞ but other, not necessarily finite dimensional, is continuous.

Proof. If we define 𝜂_𝑥(𝑡)=𝑇^′𝜇_𝑥(𝑡),𝜈_Λ𝑥(𝑡), (6.22) where 𝑇^′≫𝑇, then (𝑋,𝜂,𝑇) is an RN-space because (RN1) and (RN2) are immediate from definition, for the triangle inequality (RN3), 𝑇𝜂_𝑥(𝑡),𝜂_𝑧(𝑠)=𝑇𝑇^′𝜇_𝑥(𝑡),𝜈_Λ𝑥(𝑡),𝑇^′𝜇_𝑧(𝑠),𝜈_Λ𝑧(𝑠)≤𝑇^′𝑇𝜇_𝑥(𝑡),𝜇_𝑧(𝑠)𝑇𝜈_Λ𝑥(𝑡),𝜈_Λ𝑧(𝑠)≤𝑇^′𝜇_𝑥+𝑧(𝑡+𝑠),𝜈_{Λ(𝑥+𝑧)}(𝑡+𝑠)=𝜂_𝑥+𝑧(𝑡+𝑠). (6.23) Now, let 𝑥→−_𝑛^𝜇𝑥, then, by Theorem 6.3, 𝑥→−_𝑛^𝜂𝑥 but since, by (6.15), 𝜈_Λ𝑥(𝑡)≥𝜂_𝑥(𝑡), (6.24) then Λ𝑥→−_𝑛^𝜈Λ𝑥. Hence Λ is continuous.

Corollary 6.10. Every linear isomorphism between finite dimensional RN-spaces is topological isomorphism.

Corollary 6.11. Every finite dimensional RN-space (𝑋,𝜇,𝑇) is complete.

Proof. By Corollary 6.10, (𝑋,𝜇,𝑇) and (ℝ^𝑛,Φ,𝑇) are random topologically isomorphic. Since (ℝ^𝑛,Φ,𝑇) is complete and random topological isomorphism preserves completeness, (𝑋,𝜇,𝑇) is complete.

Definition 6.12. Let (𝑉,𝜇,𝑇) be an RN-space, 𝑊 be a linear manifold in 𝑉 and 𝑄∶𝑉→𝑉/𝑊 be the natural mapping, 𝑄𝑥=𝑥+𝑊. For any 𝑡>0, we define 𝜇(𝑥+𝑊,𝑡)=sup𝜇_𝑥+𝑦(𝑡)∶𝑦∈𝑊. (6.25)

Theorem 6.13. Let 𝑊 be a closed subspace of an RN-space (𝑉,𝜇,𝑇). If 𝑥∈𝑉 and 𝜖>0, then there is 𝑥′ in 𝑉 such that 𝑥′+𝑊=𝑥+𝑊, 𝐸_𝜆,𝜇(𝑥′)<𝐸_𝜆,𝜇(𝑥+𝑊)+𝜖.

Proof. By the properties of sup, there always exists 𝑦∈𝑊 such that 𝐸_𝜆,𝒫(𝑥+𝑦)<𝐸_𝜆,𝜇(𝑥+𝑊)+𝜖. Now, it is enough to put 𝑥′=𝑥+𝑦.

Theorem 6.14. Let 𝑊 be a closed subspace of RN-space (𝑉,𝜇,𝑇) and 𝜇 be given in the above definition. Then
(1)𝜇 is an RN-space on 𝑉/𝑊,
(2)𝜇_𝑄𝑥(𝑡)≥𝜇_𝑥(𝑡),
(3) if (𝑉,𝜇,𝑇) is a random Banach space, then so is (𝑉/𝑊,𝜇,𝑇).

Proof. It is clear that 𝜇_𝑥+𝑊(𝑡)>0. Let 𝜇_𝑥+𝑊(𝑡)=1. By definition, there is a sequence {𝑥_𝑛} in 𝑊 such that 𝜇_{𝑥+𝑥𝑛}(𝑡)→1. Thus 𝑥+𝑥_𝑛→0 or, equivalently, 𝑥_𝑛→(−𝑥) and since 𝑊 is closed, 𝑥∈𝑊 and 𝑥+𝑊=𝑊, the zero element of 𝑉/𝑊. Now, we have 𝜇_{(𝑥+𝑊)+(𝑦+𝑊)}(𝑡)=𝜇_{(𝑥+𝑦)+𝑊}(𝑡)≥𝜇_{(𝑥+𝑚)+(𝑦+𝑛)}(𝑡)≥𝑇𝜇_𝑥+𝑚𝑡₁,𝜇_𝑦+𝑛𝑡₂ (6.26) for 𝑚,𝑛∈𝑊, 𝑥,𝑦∈𝑉, and 𝑡₁+𝑡₂=𝑡. Now, if we take the sup, then we have 𝜇_{(𝑥+𝑊)+(𝑦+𝑊)}(𝑡)≥𝑇𝜇_𝑥+𝑊𝑡₁,𝜇_𝑦+𝑊𝑡₂. (6.27) Therefore, 𝜇 is random norm on 𝑉/𝑊.
(2) By Definition 6.12, we have 𝜇_𝑄𝑥(𝑡)=𝜇_𝑥+𝑊(𝑡)=sup𝜇_𝑥+𝑦(𝑡)∶𝑦∈𝑊≥𝜇_𝑥(𝑡). (6.28) Note that, by Lemma 5.14, 𝐸_𝜆,𝜇(𝑄𝑥)=inf𝑡>0∶𝜇_𝑄𝑥(𝑡)>1−𝜆≤inf𝑡>0∶𝜇_𝑥(𝑡)>1−𝜆=𝐸_𝜆,𝜇(𝑥). (6.29)
(3) Let {𝑥_𝑛+𝑊} be a Cauchy sequence in 𝑉/𝑊. Then there exists 𝑛₀∈ℕ such that, for every 𝑛≥𝑛₀, 𝐸_𝜆,𝜇((𝑥_𝑛+𝑊)−(𝑥_𝑛+1+𝑊))≤2^−𝑛. Let 𝑦₁=0. Choose 𝑦₂∈𝑊 such that 𝐸_𝜆,𝜇𝑥₁−𝑥₂−𝑦₂,𝑡≤𝐸_𝜆,𝜇𝑥₁−𝑥₂+1+𝑊2. (6.30) However, 𝐸_𝜆,𝜇((𝑥₁−𝑥₂)+𝑊)≤1/2 and so 𝐸_𝜆,𝜇(𝑥₁−(𝑥₂−𝑦₂))≤1/2².
Now, suppose that 𝑦_𝑛−1 has been chosen and so choose 𝑦_𝑛∈𝑊 such that 𝐸_𝜆,𝜇𝑥_𝑛−1+𝑦_𝑛−1−𝑥_𝑛+𝑦_𝑛≤𝐸_𝜆,𝜇𝑥_𝑛−1−𝑥_𝑛+𝑊+2^−𝑛+1. (6.31) Hence we have 𝐸_𝜆,𝜇𝑥_𝑛−1+𝑦_𝑛−1−𝑥_𝑛+𝑦_𝑛≤2^−𝑛+2. (6.32)
However, for every positive integer 𝑚>𝑛 and 𝜆∈(0,1), by Lemma 5.14, there exists 𝛾∈(0,1) such that 𝐸_𝜆,𝜇𝑥_𝑚+𝑦_𝑚−𝑥_𝑛+𝑦_𝑛≤𝐸_𝛾,𝜇𝑥_𝑛+1+𝑦_𝑛+1−𝑥_𝑛+𝑦_𝑛+⋯+𝐸_𝛾,𝜇𝑥_𝑚+𝑦_𝑚−𝑥_𝑚−1+𝑦_𝑚−1≤^𝑚_𝑖=𝑛2^−𝑖. (6.33) By Lemma 5.14, {𝑥_𝑛+𝑦_𝑛} is a Cauchy sequence in 𝑉. Since 𝑉 is complete, there is 𝑥₀ in 𝑉 such that 𝑥_𝑛+𝑦_𝑛→𝑥₀ in 𝑉. On the other hand, 𝑥_𝑛+𝑊=𝑄𝑥_𝑛+𝑦_𝑛⟶𝑄𝑥₀=𝑥₀+𝑊. (6.34) Therefore, every Cauchy sequence {𝑥_𝑛+𝑊} is convergent in 𝑉/𝑊 and so 𝑉/𝑊 is complete. Thus (𝑉/𝑊,𝜇,𝑇) is a random Banach space.

Theorem 6.15. Let 𝑊 be a closed subspace of an RN-space (𝑉,𝜇,𝑇). If two of the spaces 𝑉, 𝑊, and 𝑉/𝑊 are complete, then so is the third one.

Proof. If 𝑉 is a random Banach space, then so are 𝑉/𝑊 and 𝑊. Hence all that needs to be checked is that 𝑉 is complete whenever both 𝑊 and 𝑉/𝑊 are complete. Suppose that 𝑊 and 𝑉/𝑊 are random Banach spaces and let {𝑥_𝑛} be a Cauchy sequence in 𝑉. Since 𝐸_𝜆,𝜇𝑥_𝑛−𝑥_𝑚+𝑊≤𝐸_𝜆,𝜇𝑥_𝑛−𝑥_𝑚 (6.35) for each 𝑚,𝑛∈ℕ, the sequence {𝑥_𝑛+W} is a Cauchy sequence in 𝑉/𝑊 and so it converges to 𝑦+𝑊 for some 𝑦∈𝑊. Thus there is 𝑛₀∈ℕ such that, for every 𝑛≥𝑛₀, 𝐸_𝜆,𝜇𝑥_𝑛−𝑦+𝑊<2^−𝑛. (6.36) Now, by the last theorem, there exist a sequence {𝑦_𝑛} in 𝑉 such that 𝑦_𝑛+𝑊=(𝑥_𝑛−𝑦)+𝑊 and 𝐸_𝜆,𝜇𝑦_𝑛<𝐸_𝜆,𝜇𝑥_𝑛−𝑦+𝑊+2^−𝑛. (6.37) Thus we have lim_𝑛𝐸_𝜆,𝜇𝑦_𝑛≤0 (6.38) and, by Lemma 5.14, 𝜇_𝑦𝑛(𝑡)→1 for every 𝑡>0, that is, lim_𝑛𝑦_𝑛=0. Therefore, {𝑥_𝑛−𝑦_𝑛−𝑦} is a Cauchy sequence in 𝑊 and thus is convergent to a point 𝑧∈𝑊. This implies that {𝑥_𝑛} converges to 𝑧+𝑦 and hence 𝑉 is complete.

Theorem 6.16 (open mapping theorem). If 𝑇 is a random bounded linear operator from an RN-space (𝑉,𝜇,𝑇) onto an RN-space (𝑉^′,𝜈,𝑇), then 𝑇 is an open mapping.

Proof. The theorem will be proved in several steps.
Step 1. Let 𝐸 be a neighborhood of the 0 in 𝑉. We show 0∈(𝑇(𝐸))^𝑜. Let 𝑊 be a balanced neighborhood of 0 such that 𝑊+𝑊⊂𝐸. Since 𝑇(𝑉)=𝑉′ and 𝑊 is absorbing, it follows that 𝑉′=∩_𝑛𝑇(𝑛𝑊) and so there exists an 𝑛₀∈ℕ such that 𝑇(𝑛₀𝑊) has nonempty interior. Therefore, 0∈(𝑇(𝑊))^𝑜−(𝑇(𝑊))^𝑜. On the other hand, 𝑇(𝑊)^𝑜−𝑇(𝑊)^𝑜⊂𝑇(𝑊)−𝑇(𝑊)=𝑇(𝑊)(𝑊)⊂+𝑇𝑇(𝐸). (6.39) So, the set 𝑇(𝐸) includes the neighborhood (𝑇(𝑊))^𝑜−(𝑇(𝑊))^𝑜 of 0.
Step 2. We show 0∈(𝑇(𝐸))^𝑜. Since 0∈𝐸 and 𝐸 is an open set, there exist 0<𝛼<1 and 𝑡₀∈(0,∞) such that 𝐵₀(𝛼,𝑡₀)⊂𝐸. However, 0<𝛼<1 and so a sequence {𝜖_𝑛} can be found such that 𝑇^𝑚−𝑛1−𝜖_𝑛+1,…,1−𝜖_𝑚⟶1,1−𝛼<lim_𝑛𝑇^𝑛−11−𝜖₁,1−𝜖_𝑛, (6.40) in which 𝑚>𝑛. On the other hand, 0∈𝑇(𝐵₀(𝜖_𝑛,𝑡^′_𝑛)), where 𝑡^′_𝑛=(1/2^𝑛)𝑡₀ and so, by Step 1, there exist 0<𝜎_𝑛<1 and 𝑡_𝑛>0 such that 𝐵₀𝜎_𝑛,𝑡_𝑛⊂𝑇𝐵₀𝜖_𝑛,𝑡^′_𝑛. (6.41) Since the set {𝐵₀(𝑟,1/𝑛)} is a countable local base at zero and 𝑡^′_𝑛→0 as 𝑛→∞, so 𝑡_𝑛 and 𝜎_𝑛 can be chosen such that 𝑡_𝑛→0 and 𝜎_𝑛→0 as 𝑛→∞.
Now, we show that 𝐵₀𝜎₁,𝑡₁⊂(𝑇(𝐸))^𝑜. (6.42) Suppose 𝑦₀∈𝐵₀(,𝜎₁,𝑡₁). Then 𝑦₀∈𝑇(𝐵₀(𝜖₁,𝑡^′₁)) and so for 0<𝜎₂ and 𝑡₂>0 the ball 𝐵_𝑦0(𝜎₂,𝑡₂) intersects 𝑇(𝐵₀(𝜖₁,𝑡^′₁)). Therefore, there exists 𝑥₁∈𝐵₀(𝜖₁,𝑡^′₁) such that 𝑇𝑥₁∈𝐵_𝑦0(𝜎₂,𝑡₂), that is, 𝜈_{𝑦0−𝑇𝑥1}𝑡₂>1−𝜎₂ (6.43) or equivalently 𝑦₀−𝑇𝑥₁∈𝐵₀𝜎₂,𝑡₂⊂𝑇𝐵₀𝜖₁,𝑡^′₁. (6.44) By the similar argument, there exist 𝑥₂ in 𝐵₀(𝜖₂,𝑡^′₂) such that 𝜈_{𝑦0−(𝑇𝑥1+𝑇𝑥2)}𝑡₃=𝜈_{(𝑦0−𝑇𝑥1)−𝑇𝑥2}𝑡₃>1−𝜎₃. (6.45) If this process is continued, then it leads to a sequence {𝑥_𝑛} such that 𝑥_𝑛∈𝐵₀(𝜖_𝑛,𝑡^′_𝑛) and 𝜈_𝑦0−∑^𝑛−1_{𝑗=1𝑇𝑥𝑗}𝑡_𝑛>1−𝜎_𝑛. (6.46) Now, if 𝑛,𝑚∈ℕ and 𝑚>𝑛, then we have 𝜇_∑^𝑛_{𝑗=1𝑥𝑗−∑}^𝑚_{𝑗=𝑛+1𝑥𝑗}(𝑡)=𝜇_∑^𝑚_{𝑗=𝑛+1𝑥𝑗}(𝑡)≥𝑇^𝑚−𝑛𝜇_𝑥𝑛+1𝑡_𝑛+1,𝜇_𝑥𝑚𝑡_𝑚, (6.47) where 𝑡_𝑛+1+𝑡_𝑛+2+⋯+𝑡_𝑚=𝑡. Put 𝑡^′₀=min{𝑡_𝑛+1,𝑡_𝑛+2,…,𝑡_𝑚}. Since 𝑡^′_𝑛→0, there exists 𝑛₀∈ℕ such that 0<𝑡^′_𝑛≤𝑡^′₀ for 𝑛>𝑛₀. Therefore, for 𝑚>𝑛 we have 𝑇^𝑚−𝑛𝜇_𝑥𝑛+1𝑡^′₀,𝜇_𝑥𝑚𝑡^′₀≥𝑇^𝑚−𝑛𝜇_𝑥𝑛+1𝑡^′_𝑛+1,𝜇_𝑥𝑚𝑡^′_𝑚≥𝑇^𝑚−𝑛1−𝜖_𝑛+1,1−𝜖_𝑚. (6.48) Hence it follows that lim_𝑛→∞𝜇_∑^𝑚_{𝑗=𝑛+1𝑥𝑗}(𝑡)≥lim_𝑛→∞𝑇^𝑚−𝑛1−𝜖_𝑛+1,1−𝜖_𝑚=1. (6.49) That is, 𝜇_∑^𝑚_{𝑗=𝑛+1𝑥𝑗}(𝑡)⟶1 (6.50) for all 𝑡>0. Thus the sequence {∑^𝑛_𝑗=1𝑥_𝑗} is a Cauchy sequence and consequently the series {∑^∞_𝑗=1𝑥_𝑗} converges to some point 𝑥₀∈𝑉 because 𝑉 is a complete space.
By fixing 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>𝑡_𝑛 for 𝑛>𝑛₀ because 𝑡_𝑛→0. Thus 𝜈_{𝑦0−𝑇(∑}^𝑛−1_{𝑗=1𝑥𝑗)}(𝑡)≥𝜈_{𝑦0−𝑇(∑}^𝑛−1_{𝑗=1𝑥𝑗)}𝑡_𝑛≥1−𝜎_𝑛 (6.51) and thus 𝜈_{𝑦0−𝑇(∑}^𝑛−1_{𝑗=1𝑥𝑗)}(𝑡)⟶1. (6.52) Therefore, we have 𝑦₀=lim_𝑛𝑇^𝑛−1_𝑗=1𝑥_𝑗=𝑇lim_𝑛^𝑛−1_𝑗=1𝑥_𝑗=𝑇𝑥₀. (6.53) But it follows that 𝜇_𝑥0𝑡₀=lim_𝑛𝜇_∑^𝑛_{𝑗=1𝑥𝑗}𝑡₀≥𝑇^𝑛lim_𝑛𝜇_𝑥1𝑡^′₁,𝜇_𝑥𝑛𝑡^′_𝑛≥lim_𝑛𝑇^𝑛−11−𝜖₁,…,1−𝜖_𝑛>1−𝛼. (6.54) Hence 𝑥₀∈𝐵₀(𝛼,𝑡₀).
Step 3. Let 𝐺 be an open subset of 𝑉 and 𝑥∈𝐺. Then we have 𝑇(𝐺)=𝑇𝑥+𝑇(−𝑥+𝐺)⊃𝑇𝑥+(𝑇(−𝑥+𝐺))^𝑜. (6.55) Hence 𝑇(𝐺) is open, because it includes a neighborhood of each of its point.

Corollary 6.17. Every one-to-one random bounded linear operator from a random Banach space onto random Banach space has a random bounded inverse.

Theorem 6.18 (closed graph theorem). Let 𝑇 be a linear operator from the random Banach space (𝑉,𝜇,𝑇) into the random Banach space (𝑉′,𝜈,𝑇). Suppose that, for every sequence {𝑥_𝑛} in 𝑉 such that 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦 for some elements 𝑥∈𝑉 and 𝑦∈𝑉′, it follows that 𝑇𝑥=𝑦. Then 𝑇 is random bounded.

Proof. For any 𝑡>0, 𝑥∈𝑋 and 𝑦∈𝑉′, define Φ_(𝑥,𝑦)(𝑡)=𝑇^′𝜇_𝑥(𝑡),𝜈_𝑦(𝑡), (6.56) where 𝑇′≫𝑇.
First we show that (𝑉×𝑉′,Φ,𝑇) is a complete RN-space. The properties of (RN1) and (RN2) are immediate from the definition. For the triangle inequality (RN3), suppose that 𝑥,𝑧∈𝑉, 𝑦,𝑢∈𝑉^′, and 𝑡,𝑠>0, then 𝑇Φ_(𝑥,𝑦)(𝑡),Φ_(𝑧,𝑢)(𝑠)=𝑇𝑇^′𝜇_𝑥(𝑡),𝜈_𝑦(𝑡),𝑇^′𝜇_𝑧(𝑠),𝜈_𝑢(𝑠)≤𝑇^′𝑇𝜇_𝑥(𝑡),𝜇_𝑧(𝑠),𝑇𝜈_𝑦(𝑡),𝜈_𝑢(𝑠)≤𝑇^′𝜇_𝑥+𝑧(𝑡+𝑠),𝜈_𝑦+𝑢(𝑡+𝑠)=Φ_{(𝑥+𝑧,𝑦+𝑢)}(𝑡+𝑠). (6.57)
Now, if {(𝑥_𝑛,𝑦_𝑛)} is a Cauchy sequence in 𝑉×𝑉^′, then for every 𝜖>0 and 𝑡>0 there exists 𝑛₀∈ℕ such that Φ_{(𝑥𝑛,𝑦𝑛)−(𝑥𝑚,𝑦𝑚)}(𝑡)>1−𝜖 (6.58) for 𝑚,𝑛>𝑛₀. Thus for 𝑚,𝑛>𝑛₀, 𝑇^′𝜇_{𝑥𝑛−𝑥𝑚}(𝑡),𝜈_{𝑦𝑛−𝑦𝑚}(𝑡)=Φ_{(𝑥𝑛−𝑥𝑚,𝑦𝑛−𝑦𝑚)}(𝑡)=Φ_{(𝑥𝑛,𝑦𝑛)−(𝑥𝑚,𝑦𝑚)}(𝑡)>1−𝜖. (6.59) Therefore, {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in 𝑉 and 𝑉^′, respectively, and there exist 𝑥∈𝑉 and 𝑦∈𝑉^′ such that 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 and consequently (𝑥_𝑛,𝑦_𝑛)→(𝑥,𝑦). Hence (𝑉×𝑉^′,Φ,𝑇) is a complete RN-space. The remainder of the proof is the same as the classical case.

7. Fuzzy Normed Spaces

Now, we define the fuzzy normed spaces and give an example of these spaces. Here the 𝑡-norms notation is showed by ∗.

Definition 7.1. The triple (𝑋,𝑀,∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous 𝑡-norm, and 𝑀 is a fuzzy set on 𝑋²×(0,∞) satisfying the following conditions: for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0,
(FM1) 𝑀(𝑥,𝑦,0)>0;
(FM2) 𝑀(𝑥,𝑦,𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝑦;
(FM3) 𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡);
(FM4) 𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠) for all 𝑡,𝑠>0;
(FM5) 𝑀(𝑥,𝑦,⋅)∶(0,∞)→[0,1] is continuous.

Definition 7.2. The triple (𝑋,𝑁,∗) is said to be a fuzzy normed space if 𝑋 is a vector space, ∗ is a continuous 𝑡-norm, and 𝑁 is a fuzzy set on 𝑋×(0,∞) satisfying the following conditions: for every 𝑥,𝑦∈𝑋 and 𝑡,𝑠>0;
(FN1) 𝑁(𝑥,𝑡)>0;
(FN2) 𝑁(𝑥,𝑡)=1 if and only if 𝑥=0;
(FN3) 𝑁(𝛼𝑥,𝑡)=𝑁(𝑥,𝑡/|𝛼|), for all 𝛼≠0;
(FN4) 𝑁(𝑥,𝑡)∗𝑁(𝑦,𝑠)≤𝑁(𝑥+𝑦,𝑡+𝑠);
(FN5) 𝑁(𝑥,⋅)∶(0,∞)→[0,1] is continuous;
(FN6) lim_𝑡→∞𝑁(𝑥,𝑡)=1.

Lemma 7.3. Let 𝑁 be a fuzzy norm, then we have
(i)𝑁(𝑥,𝑡) is nondecreasing with respect to 𝑡 for each 𝑥∈𝑋,
(ii)𝑁(𝑥−𝑦,𝑡)=𝑁(𝑦−𝑥,𝑡).

Proof. Let 𝑡<𝑠 then 𝑘=𝑠−𝑡>0 and we have 𝑁()()()()()𝑥,𝑡=𝑁𝑥,𝑡∗1=𝑁𝑥,𝑡∗𝑁0,𝑘≤𝑁𝑥,𝑠 (7.1) and this proves (i).
To prove (ii) we have 𝑁()𝑡𝑥−𝑦,𝑡=𝑁−1𝑦−𝑥,𝑡=𝑁𝑦−𝑥,||.−1=𝑁𝑦−𝑥,𝑡 (7.2)

Example 7.4. Let (𝑥,‖⋅‖) be a normed space, define 𝑎∗𝑏=𝑎𝑏 or 𝑎∗𝑏=min(𝑎,𝑏) and 𝑁(𝑥,𝑡)=𝑘𝑡^𝑛𝑘𝑡^𝑛+𝑚‖𝑥‖ (7.3) for all 𝑘,𝑚,𝑛∈ℝ⁺. Then (𝑋,𝑁,∗) is a fuzzy normed space. In particular, if 𝑘=𝑛=𝑚=1, then we have 𝑁()=𝑡𝑥,𝑡‖𝑥‖,𝑡+ (7.4) which is called the standard fuzzy norm induced by norm ‖⋅‖.

Lemma 7.5. Let (𝑋,𝑁,∗) be a fuzzy normed space. If one defines 𝑀𝑥,𝑦,𝑡=𝑁𝑥−𝑦,𝑡, (7.5) then 𝑀 is a fuzzy metric on 𝑋, which is called the fuzzy metric induced by the fuzzy norm 𝑁.

We can see both definition and properties fuzzy normed spaces are very similar to random normed spaces. Then 𝑋 equipped with 𝜇_𝑥(𝑡)=𝑁(𝑥,𝑡) and 𝑇=∗ can be regarded as an RN-space. Now, we extend the definition of fuzzy metric space. In fact we extend the range of fuzzy sets to arbitrary lattice.

Definition 7.6. The triple (𝑋,𝒫,𝒯) is said to be an ℒ-fuzzy normed space (briefly, ℒ𝐹-normed space) if 𝑋 is a vector space, 𝒯 is a continuous 𝑡-norm on ℒ, and 𝒫 is an ℒ-fuzzy set on 𝑋×]0,+∞[ satisfying the following conditions: for every 𝑥,𝑦 in 𝑋 and 𝑡,𝑠 in ]0,+∞[;
(a) 𝒫(𝑥,𝑡)>_𝐿0_ℒ;
(b) 𝒫(𝑥,𝑡)=1_ℒ if and only if 𝑥=0;
(c) 𝒫(𝛼𝑥,𝑡)=𝒫(𝑥,𝑡/|𝛼|) for each 𝛼≠0;
(d) 𝒯(𝒫(𝑥,𝑡),𝒫(𝑦,𝑠))≤_𝐿𝒫(𝑥+𝑦,𝑡+𝑠);
(e) 𝒫(𝑥,⋅)∶]0,∞[→𝐿 is continuous;
(f) lim_𝑡→∞𝒫(𝑥,𝑡)=1_ℒ.

In this case, 𝒫 is called an ℒ-fuzzy norm (briefly, ℒ𝐹-norm). If 𝒫=𝒫_𝜇,𝜈 is an intuitionistic fuzzy set and the 𝑡-norm 𝒯 is t-representable then the triple (𝑋,𝒫_𝜇,𝜈,𝒯) is said to be an intuitionistic fuzzy normed space (briefly, 𝐼𝐹-normed space).

Example 7.7. Let (𝑋,‖⋅‖) be a normed space. Denote 𝒯(𝑎,𝑏)=(𝑎₁𝑏₁,min(𝑎₂+𝑏₂,1)) for all 𝑎=(𝑎₁,𝑎₂),𝑏=(𝑏₁,𝑏₂)∈𝐿^∗ and let 𝑀 and 𝑁 be fuzzy sets on 𝑋×(0,∞) defined as follows: 𝒫_𝑀,𝑁(𝑥,𝑡)=ℎ𝑡^𝑛ℎ𝑡^𝑛‖𝑥‖,𝑚‖𝑥‖+𝑚ℎ𝑡^𝑛+𝑚‖𝑥‖ (7.6) for all 𝑡,ℎ,𝑚,𝑛∈ℝ⁺. Then (𝑋,𝒫_𝑀,𝑁,𝒯) is an 𝐼𝐹-normed space.