Fuzzy pseudo-norms and fuzzy F-spaces

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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss

Fuzzy pseudo-norms and fuzzy F-spaces Sorin N˘ad˘aban Aurel Vlaicu University of Arad, Department of Mathematics and Computer Science, Str. Elena Dr˘agoi 2, RO-310330 Arad, Romania Received 25 June 2013; received in revised form 10 December 2014; accepted 20 December 2014

Abstract In the present paper we firstly introduce the notion of fuzzy pseudo-norm, then we extend, improve and complete the results obtained by T. Bag and S.K. Samanta for fuzzy norms, in the fuzzy pseudo-norms context. Lastly, we introduce and discuss the notions of fuzzy F-norm and fuzzy F-space. By means of several auxiliary results, we obtain a characterization of metrizable topological linear spaces in terms of fuzzy F-norm. © 2014 Elsevier B.V. All rights reserved. Keywords: Fuzzy pseudo-norm; Fuzzy F-norm; Fuzzy metric space; Fuzzy metrizable topological linear space

1. Introduction The models we work with, mathematical in their nature, must arrange themselves into pre-existing structures. In functional analysis, the fundamental structure is that of a topological linear space, but its degree of generality is much too high. Consequently, many of the important results in functional analysis have been obtained on Banach spaces (complete normed linear spaces). A significant number of familiar and useful topological linear spaces have a natural metric structure and are complete. Nevertheless, this metric does not come from a norm. These are Fréchet spaces, a term introduced by S. Banach in honour of M. Fréchet. Today, the term Fréchet space is used for a particular class of metrizable topological linear spaces, namely for the locally convex ones, while the term F- space is used for complete metrizable topological linear spaces. The topology of an F-space can be given by means of an F-norm. The foundations of fuzzy functional analysis were laid by A.K. Katsaras, who studied fuzzy topological linear spaces in his works [10,11]. Moreover, A.K. Katsaras was the first to introduce the notion of fuzzy norm – of a Minkowski type – on a linear space, associated to an absolutely convex (convex and balanced) absorbing fuzzy set. From A.K. Katsaras onward, many mathematicians have proposed several notions of fuzzy norm from different points of view. Thus, in 1992, C. Felbin [5] introduced the idea of fuzzy norm on a linear space by assigning a fuzzy real number to each element of the linear space. In 2003, following S.C. Cheng and J.N. Mordeson [4], T. Bag and S.K. Samanta [2] proposed another concept of fuzzy norm. The Bag–Samanta fuzzy norm type has proved to be the most adequate of all, even though it can be still polished, simplified, improved or generalized (see [1,7,14]). We must E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2014.12.010 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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note that T. Bag and S.K. Samanta, in paper [3], have continued the study of fuzzy norms, obtaining relations between fuzzy norms on one hand, and the associated family of α-norms, on the other hand. They introduced different types of fuzzy continuity and fuzzy boundedness of linear operators. In addition to this, T. Bag and S.K. Samanta have established some principles of functional analysis in fuzzy settings, which represent a foundation for the development of fuzzy functional analysis, based on their approach. However, the notion of Šerstnev space led to a concept of fuzzy norm recently introduced by I. Gole¸t [7], in 2010. Gole¸t’s notion generalizes to continuous t-norm the concept of fuzzy norm defined by S.C. Cheng and J.N. Mordeson [4]. In 2010, C. Alegre and S. Romaguera [1] proposed the term of fuzzy quasi-norm on a real linear space and obtained characterizations of metrizable topological linear spaces. Motivated by the works of T. Bag and S.K. Samanta on fuzzy normed linear spaces, the aim of this paper is to introduce the notion of fuzzy pseudo-norm and to extend and complete the results obtained by T. Bag and S.K. Samanta, for fuzzy norms, in the context of fuzzy pseudo-norms. Finally, the concept of fuzzy F-norm is introduced and used for characterization of metrizable topological linear spaces, by replacing fuzzy norms of type (N, ∗L) or (N, ·) as stated in [1], with fuzzy F-norms (F, min). By introducing the concept of fuzzy F-space, we will be able to obtain, in a further paper, fuzzy versions for classical principles of functional analysis in this much more general context, extending T. Bag’s and S.K. Samanta’s results in paper [3], as well as the theorems obtained by I. Sadeqi and F.S. Kia [17] in 2009. The structure of the paper is as follows: after the preliminary section, in Section 3, the concept of fuzzy pseudonorm is introduced. Some of the conditions of Bag–Samanta’s notion are modified (N6 and N7) or generalized (N3). Thus, any pseudo-norm induces, in a natural way, a fuzzy pseudo-norm (Example 3.3). Some decomposition theorems for fuzzy pseudo-norms into a family of pseudo-norms are obtained. The convergence in fuzzy pseudo-normed linear spaces is studied in Section 4. The concept of fuzzy F-norm is introduced in Section 5. We establish some decomposition theorems for fuzzy F-norms. In Theorem 5.8 we prove that a topological linear space is fuzzy metrizable if and only if it is metrizable, result obtained by V. Gregori and S. Romaguera [8] in 2000, without assuming a linear space structure. Also Theorem 5.8 gives a characterization of metrizable topological linear spaces. 2. Preliminaries Definition 2.1. (See [13].) The pair (X, M) is said to be a fuzzy metric space if X is an arbitrary set and M is a fuzzy set in X × X × [0, ∞) satisfying the following conditions: (M1) (M2) (M3) (M4) (M5)

M(x, y, 0) = 0, (∀)x, y ∈ X; (∀)x, y ∈ X, x = y if and only if M(x, y, t) = 1 for all t > 0; M(x, y, t) = M(y, x, t), (∀)x, y ∈ X, (∀)t > 0; M(x, z, t + s) ≥ min{M(x, y, t), M(y, z, s)}, (∀)x, y, z ∈ X, (∀)t, s > 0; (∀)x, y ∈ X, M(x, y, ·) : [0, ∞) → [0, 1] is left continuous and limt→∞ M(x, y, t) = 1.

Remark 2.2. We note that the notion of a fuzzy metric, as given by I. Kramosil and J. Michálek, is more general that the one given in Definition 2.1. In fact, the condition (M4) is M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s),

(∀)x, y, z ∈ X, (∀)t, s > 0,

where ∗ is a continuous t-norm (see [19]). The basic examples of continuous t-norms are minimum, usual multiplication denoted by · and the Lukasiewicz t-norm ∗L defined by a ∗L b = max{a + b − 1, 0}, but, in this paper we will work only with ∗ = min. Our basic reference for fuzzy metric space and related structures is [9], while for t-norms, is [12]. Definition 2.3. (See [2].) Let X be a linear space over a field K (where K is the space of real numbers R or the space of complex numbers C). A fuzzy set N in X × R is called a fuzzy norm on X if it satisfies: (N1) N(x, t) = 0, (∀)t ≤ 0; (N2) N (x, t) = 1, (∀)t ∈ R∗+ if and only if x = 0, where R∗+ denotes the set of all positive reals;

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t (N3) N (λx, t) = N(x, |λ| ), (∀)t ∈ R∗+ , (∀)λ ∈ K∗ ; (N4) N (x + y, t + s) ≥ min{N (x, t), N (y, s)}, (∀)x, y ∈ X, (∀)t, s ∈ R; (N5) limt→∞ N (x, t) = 1, (∀)x ∈ X.

The pair (X, N ) is called fuzzy normed linear space. Remark 2.4. a) We note that in the definition of fuzzy norm T. Bag and S.K. Samanta have imposed the condition “N(x, ·) is non-decreasing, for all x ∈ X”, but this is a consequence of (N2) and (N4), so it should be omitted. b) In order to obtain some important results, T. Bag and S.K. Samanta assume that the fuzzy norm also satisfies: (N6) N(x, t) > 0, (∀)t ∈ R∗+ ⇒ x = 0; (N7) (∀)x = 0, N (x, ·) is a continuous function on R and strictly increasing on the subset {t : 0 < N (x, t) < 1} of R. c) The notion of Šerstnev space led to a concept of fuzzy norm recently introduced by I. Gole¸t [7], in 2010. Gole¸t’s notion generalizes to continuous t-norm the concept of fuzzy norm defined by S.C. Cheng and J.N. Mordeson [4]. In fact, the condition (N4) becomes N (x + y, t + s) ≥ N (x, t) ∗ N (y, s),

(∀)x, y ∈ X, (∀)t, s ∈ R.

For this type of fuzzy norms we will use the notation (N, ∗). Definition 2.5. (See [18].) A pseudo-norm on a linear space X is a function X  x → |x| ∈ R which satisfies: (PN1) (PN2) (PN3) (PN4)

|x| ≥ 0, (∀)x ∈ X; |x| = 0 if and only if x = 0; |λx| ≤ |x|, (∀)x ∈ X, (∀)λ ∈ K, |λ| ≤ 1; |x + y| ≤ |x| + |y|, (∀)x, y ∈ X.

Remark 2.6. It is clear that a pseudo-norm on X defines, via the invariant metric d(x, y) = |x − y|, a topology T on X. This topology is not compatible with the structure of linear space of X. If, in addition, the pseudo-norm satisfies two more conditions (PN5) λn → 0 ⇒ |λn x| → 0, (∀)x ∈ X, (PN6) |xn | → 0 ⇒ |λxn | → 0, (∀)λ ∈ K, then, the topology T is compatible with the structure of linear space of X and thus we obtain a topological linear space. If, in addition, the induced invariant metric d is complete, it will be called F-space (see [16, 1.8]), and a pseudo-norm which satisfies (PN5) and (PN6) will be named F-norm. 3. Fuzzy pseudo-norm Definition 3.1. Let X be a linear space over a field K (where K is R or C). A fuzzy set F in X × R is called a fuzzy pseudo-norm on X if it satisfies: (F1) (F2) (F3) (F4) (F5)

F (x, t) = 0, (∀)x ∈ X, (∀)t ≤ 0; F (x, t) = 1, (∀)t ∈ R∗+ if and only if x = 0; F (λx, t) ≥ F (x, t), (∀)x ∈ X, (∀)t ∈ R, (∀)λ ∈ K, |λ| ≤ 1; F (x + y, t + s) ≥ min{F (x, t), F (y, s)}, (∀)x, y ∈ X, (∀)t, s ∈ R; limt→∞ F (x, t) = 1, (∀)x ∈ X.

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(F6) If there exists α0 ∈ (0, 1) such that F (x, t) > α0 , (∀)t ∈ R∗+ , then x = 0; (F7) (∀)x ∈ X, F (x, ·) is left continuous on R. The pair (X, F ) will be called fuzzy pseudo-normed linear space. Remark 3.2. a) (∀)x ∈ X, F (x, ·) is non-decreasing function on R. b) It is easily seen that any fuzzy norm of the Bag–Samanta type is a fuzzy pseudo-norm. We verify (F3). For λ = 0, we have N (λx, t) = N (0, t) = 1 and then N (x, t) ≤ N (λx, t) = 1. If λ ∈ K, |λ| ≤ 1, t t t then |λ| ≥ t and as N is a non-decreasing function, we obtain that N (x, |λ| ) ≥ N (x, t). But N (x, |λ| ) = N (λx, t) and thus N(λx, t) ≥ N(x, t). We also remark that (F6) is a weaker condition than (N6) and (F7) is a weaker condition than (N7). Example 3.3. Let X be a linear space and | · | be a pseudo-norm on X. Then: 

if x ∈ X, t ∈ R, t > 0 0 if x ∈ X, t ∈ R, t ≤ 0 is a fuzzy pseudo-norm on X;  1 if |x| < t b) F (x, t) := 0 if |x| ≥ t is a fuzzy pseudo-norm on X. F (x, t) :=

a)

t t+|x|

Proof. a) (F1) It is obvious. t (F2) If F (x, t) = 1, (∀)t ∈ R∗+ , then t+|x| = 1, (∀)t > 0. Thus |x| = 0 and therefore x = 0. Conversely, if x = 0, then |x| = 0 and F (x, t) = 1, (∀)t > 0. t t (F3) F (λx, t) = ≥ = F (x, t), (∀)t > 0, (∀)λ ∈ K, |λ| ≤ 1. t + |λx| t + |x| If t = 0, then F (λx, t) = F (x, t) = 0. (F4) Let x, y ∈ X, t, s ∈ R∗+ . We will assume, without restricting the general case, that F (x, t) ≤ F (y, s). Then t s t+|x| ≤ s+|y| . Thus t + |x| + s + |y| ≤ t + |x| +

 t +s  s t + |x| = t + |x| . t t

As a consequence t +s t +s ≥ t + s + |x + y| t + s + |x| + |y|   t t +s = = F (x, t) = min F (x, t), F (y, s) . ≥ t+s t + |x| [t + |x|] t

F (x + y, t + s) =

If t ≤ 0 or s ≤ 0, then F (x, t) = 0 or F (y, s) = 0 and therefore min{F (x, t), F (y, s)} = 0. As a consequence, the inequality F (x + y, t + s) ≥ min{F (x, t), F (y, s)} is obvious. t (F5) Obviously, limt→∞ F (x, t) = limt→∞ t+|x| = 1. t (F6) We suppose that there exists α0 ∈ (0, 1) such that F (x, t) > α0 , (∀)t ∈ R∗+ . Then t+|x| > α0 , (∀)t > 0. α0 Hence t > 1−α0 |x|, (∀)t > 0. This implies that |x| = 0 and therefore x = 0. (F7) We have to verify, the left continuity of the function F (x, ·) in t = 0, otherwise being obviously continut ous. For x = 0, as limt→0,t<0 F (x, t) = 0, limt→0,t>0 F (x, t) = limt→0,t>0 t+|x| = 0 and F (x, 0) = 0 we obtain that F (x, ·) is continuous in t = 0. If x = 0, then limt→0,t<0 F (x, t) = 0, limt→0,t>0 F (x, t) = 1 and F (0, 0) = 0. Thus F (0, ·) is left continuous in t = 0.

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b) (F1) If t ≤ 0, then for all x ∈ X we have |x| ≥ t. Thus F (x, t) = 0. (F2) F (x, t) = 1, (∀)t ∈ R∗+ ⇔ |x| < t, (∀)t ∈ R∗+ ⇔ x = 0. (F3) Let x ∈ X, t ∈ R, λ ∈ K, |λ| < 1. If F (x, t) = 0, then the inequality F (λx, t) ≥ F (x, t) is obvious. If F (x, t) = 1, then |x| < t. Hence |λx| = |λ| · |x| ≤ |x| < t. Thus F (λx, t) = 1. Therefore F (λx, t) = F (x, t). (F4) Fix x, y ∈ X, t, s ∈ R. If min{F (x, t), F (y, s)} = 0, then the inequality   F (x + y, t + s) ≥ min F (x, t), F (y, s) is obvious. If min{F (x, t), F (y, s)} = 1, then F (x, t) = 1 and F (y, s) = 1. Therefore |x| < t and |y| < s. Thus |x + y| ≤ |x| + |y| < t + s. Hence F (x + y, t + s) = 1 and F (x + y, t + s) = min{F (x, t), F (y, s)}. (F5) It is obvious. (F6) We suppose that there exists α0 ∈ (0, 1) such that F (x, t) > α0 , (∀)t > 0. Hence F (x, t) = 1,

(∀)t > 0.

From (F2) we obtain that x = 0. (F7) Let x ∈ X. It is obvious that F (x, ·) is continuous on the sets     t ∈ R : t > |x| , t ∈ R : t < |x| . We have to verify the left continuity of F (x, ·) in t0 = |x|. As limt→t0 ,t 0 : F (x, t) > α , α ∈ (0, 1). Then {| · |α }α∈(0,1) is an ascending family of pseudo-norms on X and they will be called α-pseudo-norms. Proof. (PN1) It is obvious that |x|α ≥ 0,(∀)x ∈ X.  (PN2) |x|α = 0 ⇒ inf t > 0 : F (x, t) > α = 0 ⇒ F (x, t) > α, (∀)t > 0. From (F6) we have that x = 0. Conversely, if x = 0, then F (x, t) = 1, (∀)t > 0 and therefore F (x, t) > α, (∀)t > 0, (∀)α ∈ (0, 1). Thus inf{t > 0 : F (x, t) > α} = 0, (∀)α ∈ (0, 1), namely |x|α = 0, (∀)α ∈ (0, 1). (PN3) If λ ∈ K, |λ| ≤ 1, then F (λx, t) ≥ F (x, t). Thus     t > 0 : F (x, t) > α ⊆ t > 0 : F (λx, t) > α and therefore inf{t > 0 : F (λx, t) > α} ≤ inf{t > 0 : F (x, t) > α}. Hence |λx|α ≤ |x|α . (PN4)     |x|α + |y|α = inf t > 0 : F (x, t) > α + inf s > 0 : F (y, s) > α       = inf t + s : F (x, t) > α, F (y, s) > α = inf t + s : min F (x, t), F (y, s) > α   ≥ inf t + s : F (x + y, t + s) > α = |x + y|α . Finally, for α1 ≤ α2 , we have that     t > 0 : F (x, t) > α2 ⊆ t > 0 : F (x, t) > α1 and therefore inf{t > 0 : F (x, t) > α1 } ≤ inf{t > 0 : F (x, t) > α2 }. Thus |x|α1 ≤ |x|α2 . Therefore {| · |α }α∈(0,1) is an ascending family of pseudo-norms on X. 2 Proposition 3.5. Let (X, F ) be a fuzzy pseudo-normed linear space. Let   |x|α := inf t > 0 : F (x, t) > α , α ∈ (0, 1). Then

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1. F (x, |x|α ) ≤ α, (∀)α ∈ (0, 1), (∀)x ∈ X; 2. For x ∈ X, s > 0, α ∈ (0, 1), we have that [|x|α < s if and only if F (x, s) > α]. Proof. 1. As |x|α is the infimum of the set {t > 0 : F (x, t) > α}, we have that F (x, t) > α implies |x|α ≤ t . Thus t < |x|α ⇒ F (x, t) ≤ α. Therefore F (x, |x|α ) = limt→|x|α ,t<|x|α F (x, t) ≤ α. 2. “⇒” First, we remark that there exists t0 ∈ {t > 0 : F (x, t) > α} such that t0 < s. As F (x, ·) is non-decreasing, we obtain that F (x, t0 ) ≤ F (x, s). Hence F (x, s) > α. “⇐” We suppose that |x|α ≥ s. As F (x, ·) is non-decreasing, we obtain that F (x, |x|α ) ≥ F (x, s). Hence F (x, |x|α ) > α, which contradicts (1). Thus |x|α < s. 2 Definition 3.6. An ascending family {| · |α }α∈(0,1) of pseudo-norms on a linear space X is called right continuous if (∀)β ∈ (0, 1), (∀)x ∈ X, we have limα→β,α>β |x|α = |x|β . Theorem 3.7. Let (X, F ) be a fuzzy pseudo-normed linear space. Let   |x|α := inf t > 0 : F (x, t) > α , α ∈ (0, 1). Then {| · |α }α∈(0,1) is a right continuous and an ascending family of pseudo-norms on X. Proof. Based on Theorem 3.4, it remains to be verified that the family of pseudo-norms {| · |α }α∈(0,1) is right continuous. Let β ∈ (0, 1), x ∈ X. Let s > |x|β . Then F (x, s) > β. If α ∈ (β, F (x, s)), then we have that |x|β ≤ |x|α < s. As s is taken arbitrarily, we obtain that limα→β,α>β |x|α = |x|β . 2 Theorem 3.8. Let {| · |α }α∈(0,1) be a right continuous and an ascending family of pseudo-norms on a linear space X. Let F  : X × R → [0, 1] defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0  . F (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅ Then F  is a fuzzy pseudo-norm on X. Proof. (F1) If t ≤ 0, from the definition of F  , we have that F  (x, t) = 0. (F2) We suppose that F  (x, t) = 1, (∀)t > 0. Then sup{α ∈ (0, 1) : |x|α < t} = 1, (∀)t > 0. This means that, for all α ∈ (0, 1), we have |x|α < t, (∀)t > 0. Thus |x|α = 0 and therefore x = 0. Conversely, if x = 0, then |x|α = 0, (∀)α ∈ (0, 1). Thus, (∀)α ∈ (0, 1), we have that |x|α < t, (∀)t > 0. Therefore sup{α ∈ (0, 1) : |x|α < t} = 1, (∀)t > 0, namely F  (x, t) = 1, (∀)t > 0. (F3) If t ≤ 0, we have that F  (λx, t) = F  (x, t) = 0. If t > 0, λ ∈ K, |λ| ≤ 1, as |λx|α ≤ |x|α , we obtain that     α ∈ (0, 1) : |x|α < t ⊆ α ∈ (0, 1) : |λx|α < t . Therefore     sup α ∈ (0, 1) : |x|α < t ≤ sup α ∈ (0, 1) : |λx|α < t . Hence F  (x, t) ≤ F  (λx, t). (F4) First, we will prove that, for x ∈ X, α ∈ (0, 1), t > 0, we have [F  (x, t) > α ⇔ |x|α < t]. “⇒” We suppose that |x|α ≥ t . Let β ∈ (0, 1) : |x|β < t. Then β ≤ α. Hence   sup β ∈ (0, 1) : |x|β < t ≤ α, i.e. F  (x, t) ≤ α, which is a contradiction. Thus |x|α < t. “⇐” As |x|α < t, we obtain that α ∈ {β ∈ (0, 1) : |x|β < t}. Thus α ≤ sup{β ∈ (0, 1) : |x|β < t}, namely α ≤ F  (x, t). We suppose that F  (x, t) = α. Thus [|x|β < t ⇒ β ≤ α]. Hence [β > α ⇒ |x|β ≥ t]. Therefore |x|α = limβ→α,β>α |x|β ≥ t , which is a contradiction. Hence F  (x, t) > α.

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Now, we come back to (F4), we suppose that F  (x + y, t + s) < min{F  (x, t), F  (y, s)}. Then there exists α ∈ (0, 1) such that   F  (x + y, t + s) < α < min F  (x, t), F  (y, s) . As α < min{F  (x, t), F  (y, s)}, we obtain that α < F  (x, t) and α < F  (y, s). Thus |x|α < t and |y|α < s. Therefore |x + y|α ≤ |x|α + |y|α < t + s. Hence F  (x + y, t + s) ≥ α, contradiction with the fact that F  (x + y, t + s) < α. Thus   F  (x + y, t + s) ≥ min F  (x, t), F  (y, s) . (F5) Let x ∈ X. We will prove that limt→∞ F  (x, t) = 1. But lim F  (x, t) = 1

t→∞

⇒

F  (x, t) > 1 − 

⇔

(∀) > 0, (∃)t0 > 0 : t > t0

⇔

(∀) > 0, (∃)t0 > 0 : F  (x, t0 ) > 1 − 

⇔

(∀) > 0, (∃)t0 > 0 : |x|1− < t0

which is true, if we choose a big enough t0 . (F6) We suppose that (∃)α0 ∈ (0, 1) such that F  (x, t) > α0 , (∀)t > 0. Then |x|α0 < t, (∀)t > 0. Thus |x|α0 = 0, hence x = 0. (F7) Let x ∈ X. We will prove that F  (x, ·) is left continuous. Let t0 > 0. Let (tn ) an increasing sequence, convergent to t0 . Let α < F  (x, t0 ). Then |x|α < t0 . Then there exists n0 ∈ N such that |x|α < tn ≤ t0 , (∀)n ≥ n0 . Thus F  (x, tn ) > α, (∀)n ≥ n0 . As α is taken arbitrarily, we obtain that F  (x, tn ) → F  (x, t0 ). 2 Remark 3.9. Along the proof of the previous theorem, we obtain that if {| · |α }α∈(0,1) is a right continuous and an ascending family of pseudo-norms on a linear space X and F  : X × R → [0, 1] is defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0 F  (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅ then, for α ∈ (0, 1), t > 0, x ∈ X, we have |x|α < t

⇔

F  (x, t) > α.

Theorem 3.10. Let (X, F ) be a fuzzy pseudo-normed linear space. Let   |x|α := inf t > 0 : F (x, t) > α , α ∈ (0, 1) and F  : X × R → [0, 1] defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0 F  (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅. Then: 1. {| · |α }α∈(0,1) is a right continuous and an ascending family of pseudo-norms on X; 2. F  is a fuzzy pseudo-norm on X; 3. F  = F . Proof. 1) Is obtained in Theorem 3.7. 2) Is obtained in Theorem 3.8. 3)     F  (x, t) = sup α ∈ (0, 1) : |x|α < t = sup α ∈ (0, 1) : F (x, t) > α = F (x, t).

2

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Theorem 3.11. Let X be a linear space and {| · |α }α∈(0,1) be a right continuous and an ascending family of pseudonorms on X. Let F  : X × R → [0, 1] defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0 F  (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅. Let |x|α : X → R defined by   |x|α := inf t > 0 : F  (x, t) > α ,

α ∈ (0, 1).

Then: 1. F  is a fuzzy pseudo-norm on X; 2. {| · |α }α∈(0,1) is a right continuous and an ascending family of pseudo-norms on X; 3. |x|α = |x|α , (∀)x ∈ X, (∀)α ∈ (0, 1). Proof. 1) Is obtained in Theorem 3.8. 2) Is obtained in Theorem 3.7. 3) For x ∈ X, s > 0, α ∈ (0, 1), from Remark 3.9 we have that [|x|α < s ⇔ F  (x, s) > α]. Proposition 3.5 implies that [|x|α < s ⇔ F  (x, s) > α]. Thus [|x|α < s ⇔ |x|α < s]. Now we suppose that (∃)x ∈ X, α ∈ (0, 1) such that |x|α = |x|α . Then there exists s > 0 such that |x|α < s < |x|α or |x|α < s < |x|α , which contradicts the result that [|x|α < s ⇔ |x|α < s]. Therefore |x|α = |x|α , (∀)x ∈ X, (∀)α ∈ (0, 1). 2 Theorem 3.12. If (X, F ) is a fuzzy pseudo-normed linear space, then X is a fuzzy metric space. Moreover, M : X × X × [0, ∞) → [0, 1] defined by M(x, y, t) = F (x − y, t),

(∀)x, y ∈ X, (∀)t ≥ 0

is a fuzzy metric on X. Proof. (M1) M(x, y, 0) = F (x − y, 0) = 0. (M2) M(x, y, t) = 1, (∀)t > 0 ⇔ F (x − y, t) = 1, (∀)t > 0 ⇔ x − y = 0 ⇔ x = y. (M3) From (F3) we have that F (−x, t) ≥ F (x, t). Replacing x with −x, we obtain that F (x, t) ≥ F (−x, t). Hence F (−x, t) = F (x, t),

(∀)x ∈ X, (∀)t ∈ R.

Therefore F (x − y, t) = F (y − x, t), (∀)x, y ∈ X, (∀)t ∈ R. Thus M(x, y, t) = M(y, x, t),

(∀)x, y ∈ X, (∀)t > 0.

(M4)   M(x, z, t + s) = F (x − z, t + s) ≥ min F (x − y, t), F (y − z, s)   = min M(x, y, t), M(y, z, s) , (∀)x, y, z ∈ X, (∀)t, s > 0. (M5) From (F7) we have that F (x − y, ·) is left continuous on R. Thus M(x, y, ·) is left continuous. From (F5) we obtain that limt→∞ M(x, y, t) = limt→∞ F (x − y, t) = 1. 2 Corollary 3.13. Let (X, F ) be a fuzzy pseudo-normed linear space. For x ∈ X, r ∈ (0, 1), t > 0, we define   B(x, r, t) := y ∈ X : F (x − y, t) > 1 − r .

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Let

9

  TF = T ⊂ X : x ∈ T iff (∃)t > 0, r ∈ (0, 1) such that B(x, r, t) ⊂ T .

Then TF is a topology on X. Proof. The result follows from Theorem 3.12 and the fact that every fuzzy metric induces a topology (see [6]). 2 4. Convergence in fuzzy pseudo-normed linear spaces Definition 4.1. Let (X, F ) be a fuzzy pseudo-normed linear space and (xn ) be a sequence in X. The sequence (xn ) is said to be convergent if (∃)x ∈ X such that limn→∞ F (xn − x, t) = 1, (∀)t > 0. In this case, x is called the limit of the sequence (xn ) and we denote limn→∞ xn = x or xn → x. Definition 4.2. Let (X, F ) be a fuzzy pseudo-normed linear space. The sequence (xn ) is called Cauchy sequence if (∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : F (xn − xm , t) > 1 − r, (∀)n, m ≥ n0 . We note that previous definition is motivated by the work of A. George and P. Veeramani [6]. Proposition 4.3. Let (X, F ) be a fuzzy pseudo-normed linear space. A sequence (xn ) is convergent if and only if (xn ) is convergent in the topology TF . Proof. xn → x in the topology TF

⇔

(∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : xn ∈ B(x, r, t), (∀)n ≥ n0

⇔

(∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : F (xn − x, t) > 1 − r, (∀)n ≥ n0

⇔

lim F (xn − x, t) = 1, (∀)t > 0.

n→∞

2

Proposition 4.4. If (X, F ) is a fuzzy pseudo-normed linear space, then every convergent sequence is a Cauchy sequence. Proof. Let r ∈ (0, 1), t > 0. Then        t t t t F (xn − xm , t) = F xn − x + x − xm , + ≥ min F xn − x, , F x − xm , . 2 2 2 2 As (xn ) is a convergent sequence, converging to x, for r ∈ (0, 1), t > 0, (∃)n0 ∈ N such that F (xn − x, 2t ) > 1 − r, (∀)n ≥ n0 . Then, for n, m ≥ n0 , we have that      t t min F xn − x, , F x − xm , > 1 − r. 2 2 Hence F (xn − xm , t) > 1 − r, (∀)n, m ≥ n0 .

2

Definition 4.5. Let (X, F ) be a fuzzy pseudo-normed linear space. (X, F ) is said to be complete if any Cauchy sequence in X is convergent to a point in X. Definition 4.6. Let (X, F ) be a fuzzy pseudo-normed linear space, α ∈ (0, 1) and (xn ) be a sequence in X. The sequence (xn ) is said to be α-convergent if exists x ∈ X such that (∀)t > 0, (∃)n0 ∈ N : F (xn − x, t) > α, (∀)n ≥ n0 . α

In this case, x is called α-limit of the sequence (xn ) and we denote xn → x.

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Remark 4.7. It is obvious that, from the above definition, we obtain: (xn ) is α-convergent

⇒

lim F (xn − x, t) ≥ α, (∀)t > 0.

n→∞

Hence, any α-convergent sequence in the sense of Bag and Samanta (see [3]) is α-convergent as defined above. The converse is not generally true and this can be shown by examples as those in paper [3]. Remark 4.8. It is obvious that any convergent sequence (xn) is α-convergent, for all α ∈ (0, 1). Conversely, if a sequence (xn ) is α-convergent for all α ∈ (0, 1), then (xn ) is convergent. Proposition 4.9. Let (X, F ) be a fuzzy pseudo-normed linear space. A sequence (xn ) is α-convergent to a point x if and only if |xn − x|α → 0, as n → ∞. Proof. |xn − x|α → 0

⇔

(∀)t > 0, (∃)n0 ∈ N : |xn − x|α < t, (∀)n ≥ n0

⇔

(∀)t > 0, (∃)n0 ∈ N : F (xn − x, t) > α, (∀)n ≥ n0

⇔

α

xn → x.

2

Definition 4.10. Let (X, F ) be a fuzzy pseudo-normed linear space, α ∈ (0, 1) and (xn ) be a sequence in X. Sequence (xn ) is said to be α-Cauchy if (∀)t > 0, (∃)n0 ∈ N : F (xn − xm , t) > α, (∀)n, m ≥ n0 . Proposition 4.11. Let (X, F ) be a fuzzy pseudo-normed linear space and α ∈ (0, 1). Then every α-convergent sequence is α-Cauchy. Proof. Let (xn ) be an α-convergent sequence, converging to x. Then (∀)t > 0, (∃)n0 ∈ N : F (xn − x, t) > α, (∀)n ≥ n0 . Hence, for t > 0, we have that        t t t t F (xn − xm , t) = F xn − x + x − xm , + ≥ min F xn − x, , F x − xm , > α, 2 2 2 2 (∀)n, m ≥ n0 . Thus (xn ) is α-Cauchy.

2

Definition 4.12. Let (X, F ) be a fuzzy pseudo-normed linear space and α ∈ (0, 1). (X, F ) is said to be α-complete if any α-Cauchy sequence in X is α-convergent to a point in X. 5. Fuzzy F-spaces Definition 5.1. Let X be a linear space over o field K (where K is R or C). A fuzzy pseudo-norm F on X is said to be fuzzy F-norm if it satisfies (F8) λn → 0 ⇒ λn x → 0, (∀)x ∈ X; α α (F9) xn → 0 ⇒ λxn → 0, (∀)λ ∈ K. The pair (X, F ) will be called fuzzy F-normed linear space. Example 5.2. Let X be a linear space and | · | be a F-norm on X. Then: 

a)

if x ∈ X, t ∈ R, t > 0 0 if x ∈ X, t ∈ R, t ≤ 0 is a fuzzy F-norm on X. We call this fuzzy F-norm, induces by the F-norm | · |, the standard fuzzy F-norm. F (x, t) :=

t t+|x|

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b)

1 if |x| < t 0 if |x| ≥ t is a fuzzy F-norm on X. F (x, t) :=

Proof. a) (F8) If λn → 0, then |λn x| → 0. Therefore lim F (λn x, t) = 1,

n→∞

t t+|λn x|

→ 1, (∀)t > 0, namely

(∀)t > 0.

This means that λn x → 0. α (F9) If xn → 0, then (∀)t > 0, (∃)n0 ∈ N : F (xn , t) > α, (∀)n ≥ n0 . Thus t > α, (∀)n ≥ n0 t + |xn | 1−α (∀)t > 0, (∃)n0 ∈ N : |xn | < t , (∀)n ≥ n0 α

(∀)t > 0, (∃)n0 ∈ N : ⇒

Hence |λxn | → 0, (∀)λ ∈ K. Then limn→∞

t t+|λxn |

⇒

|xn | → 0.

= 1, (∀)t > 0. Thus limn→∞ F (λxn , t) = 1, (∀)t > 0. α

We obtain that λxn → 0, (∀)λ ∈ K and therefore λxn → 0, (∀)λ ∈ K. b) (F8) If λn → 0, then |λn x| → 0. Thus (∀)t > 0, (∃)n0 ∈ N : |λn x| < t, (∀)n ≥ n0 . Hence (∀)t > 0, (∃)n0 ∈ N : F (λn x, t) = 1, (∀)n ≥ n0 . Therefore limn→∞ F (λn x, t) = 1, (∀)t > 0. This means that λn x → 0. α (F9) If xn → 0, then (∀)t > 0, (∃)n0 ∈ N : F (xn , t) > α, (∀)n ≥ n0 . Thus (∀)t > 0, (∃)n0 ∈ N : F (xn , t) = 1, (∀)n ≥ n0 , namely (∀)t > 0, (∃)n0 ∈ N : |xn | < t, (∀)n ≥ n0 . Therefore |xn | → 0. Hence |λxn | → 0, (∀)λ ∈ K. Thus (∀)t > 0, (∃)n0 ∈ N : |λxn | < t, (∀)n ≥ n0 . This implies that (∀)t > 0, (∃)n0 ∈ N : F (λxn , t) = 1, (∀)n ≥ n0 . α

Thus limn→∞ F (λxn , t) = 1, (∀)t > 0. Hence λxn → 0, (∀)λ ∈ K and therefore λxn → 0, (∀)λ ∈ K.

2

Proposition 5.3. If (X, F ) is a fuzzy F-normed linear space, then the topology TF is compatible with the structure of linear space of X, namely the applications X × X  (x, y)

→

x +y ∈X

K × X  (λ, x)

→

λx ∈ X

are continuous and thus X is a topological linear space. Proof. We suppose that xn → x, yn → y. We will prove that xn + yn → x + y. This means that limn→∞ F (xn + yn − x − y, t + s) = 1, (∀)t, s > 0. But F (xn + yn − x − y, t + s) ≥ min{F (xn − x, t), F (yn − y, s)}. As limn→∞ F (xn − x, t) = 1, (∀)t > 0 and limn→∞ F (yn − y, s) = 1, (∀)s > 0 we obtain that lim F (xn + yn − x − y, t + s) = 1,

n→∞

(∀)t, s > 0.

Now, we suppose that xn → x0 , λn → λ0 . We will prove that λn xn → λ0 x0 . We note that λ n x n − λ 0 x 0 = λ n x n − λ n x 0 + λn x 0 − λ 0 x n + λ0 x n − λ 0 x 0 + λ 0 x 0 − λ0 x 0 = λn (xn − x0 ) − λ0 (xn − x0 ) + (λn − λ0 )x0 + λ0 (xn − x0 ) = (λn − λ0 )(xn − x0 ) + (λn − λ0 )x0 + λ0 (xn − x0 ). Therefore





 

F (λn xn − λ0 x0 , t1 + t2 + t3 ) ≥ min F (λn − λ0 )(xn − x0 ), t1 , F (λn − λ0 )x0 , t2 , F λ0 (xn − x0 ), t3 .

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As λn → λ0 , from (F8) we have that (λn −λ0 )x0 → 0, namely limn→∞ F ((λn −λ0 )x0 , t2 ) = 1, (∀)t2 > 0. As xn → x0 , α we have that xn → x0 , (∀)α ∈ (0, 1). Applying (F9) we obtain that α

λ0 (xn − x0 ) → 0,

(∀)α ∈ (0, 1),

which implies λ0 (xn − x0 ) → 0. Hence limn→∞ F (λ0 (xn − x0 ), t3 ) = 1, (∀)t3 > 0. On the other hand, as λn → λ0 , (∃)n0 ∈ N : |λn − λ0 | < 1. Applying (F3) we obtain that

 F (λn − λ0 )(xn − x0 ), t1 ≥ F (xn − x0 , t1 ), (∀)n ≥ n0 , (∀)t1 > 0. But limn→∞ F (xn − x0 , t1 ) = 1, (∀)t1 > 0. Thus limn→∞ F ((λn − λ0 )(xn − x0 ), t1 ) = 1, (∀)t1 > 0. Hence lim F (λn xn − λ0 x0 , t1 + t2 + t3 ) = 1,

n→∞

and therefore λn xn → λ0 x0 .

(∀)t1 , t2 , t3 > 0

2

Theorem 5.4. Let (X, F ) be a fuzzy F-normed linear space. Let   |x|α := inf t > 0 : F (x, t) > α , α ∈ (0, 1). Then {| · |α }α∈(0,1) is a right continuous and an ascending family of F-norms on X and they will be called α-F-norms. Proof. Based on Theorem 3.7, it remains to be verified (PN5) and (PN6). (PN5) Let λn → 0. Then λn x → 0, (∀)x ∈ X, namely limn→∞ F (λn x, t) = 1, (∀)t > 0. From here we have that (∀)α ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : F (λn x, t) > α, (∀)n ≥ n0 . Thus (∀)α ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : |λn x|α < t, (∀)n ≥ n0 . In conclusion, for any α ∈ (0, 1), we have |λn x|α → 0. α (PN6) We fix α ∈ (0, 1). We suppose that xn → 0 in | · |α . From Proposition 4.9, we obtain that xn → 0. From (F9) α we have that λxn → 0, (∀)λ ∈ K, and Proposition 4.9 leads to |λxn |α → 0, (∀)λ ∈ K. 2 Theorem 5.5. Let {| · |α }α∈(0,1) be a right continuous and an ascending family of F-norms on a linear space X. Let F  : X × R → [0, 1] defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0  F (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅. Then F  is a fuzzy F-norm on X. Proof. Based on Theorem 3.8, we have that F  is a fuzzy pseudo-norm on X and it remains to verify (F8) and (F9). (F8) Let λn → 0. Then, (∀)α ∈ (0, 1), we have that |λn x|α → 0. Hence (∀)α ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : |λn x|α < t, (∀)n ≥ n0 . Thus (∀)α ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : F  (λn x, t) > α, (∀)n ≥ n0 . Therefore limn→∞ F  (λn x, t) = 1, (∀)t > 0. Consequently, λn x → 0. α (F9) We suppose that xn → 0. Then (∀)t > 0, (∃)n0 ∈ N : F  (xn , t) > α, (∀)n ≥ n0 . From Remark 3.9 we obtain that (∀)t > 0, (∃)n0 ∈ N : |xn |α < t, (∀)n ≥ n0 .

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This means that |xn |α → 0. As | · |α is a F-norm on X, we have that |λxn |α → 0, (∀)λ ∈ K. But |λxn |α → 0 implies that (∀)t > 0, (∃)n0 ∈ N : |λxn |α < t, (∀)n ≥ n0 . Remark 3.9 leads to (∀)t > 0, (∃)n0 ∈ N : F  (λxn , t) > α, (∀)n ≥ n0 . α

Therefore λxn → 0.

2

Theorem 5.6. Let (X, F ) be a fuzzy F-normed linear space. Let   |x|α := inf t > 0 : F (x, t) > α , α ∈ (0, 1) and F  : X × R → [0, 1] defined by  sup{α ∈ (0, 1) : |x|α < t} if t > 0  F (x, t) = 0 if t ≤ 0 or {α ∈ (0, 1) : |x|α < t} = ∅. Then: 1. {| · |α }α∈(0,1) is a right continuous and an ascending family of F-norms on X; 2. F  is a fuzzy F-norm on X; 3. F  = F . Proof. It is an immediate consequence of the previous theorems and Theorem 3.10.

2

Proposition 5.7. Let X be a linear space and | · | be a F-norm on X. Let  t if x ∈ X, t ∈ R, t > 0 F (x, t) := t+|x| 0 if x ∈ X, t ∈ R, t ≤ 0 be the corresponding standard fuzzy F-norm on X. Then topology T induced by the F-norm | · | and topology TF induced by the standard fuzzy F-norm are the same. Proof. Let A ∈ T . We will prove that A ∈ TF . This means that (∀)x ∈ A, (∃)r ∈ (0, 1), t > 0 such that B(x, r, t) ⊂ A.  Let x ∈ A. Then there exists  > 0 such that B(x, ) ⊂ A. Let t = 1 and r = 1+ ∈ (0, 1). If y ∈ B(x, r, t), it follows that  1 F (x − y, t) > 1 − r = 1 − = . 1+ 1+ 1 1 Thus 1+|x−y| > 1+ . Therefore |x − y| < . Hence y ∈ B(x, ) ⊂ A. This shows that B(x, r, t) ⊂ A. Conversely, let A ∈ TF .We will prove that A ∈ T . This means that (∀)x ∈ A, (∃) > 0 such that B(x, ) ⊂ A. tr Let x ∈ A. As A ∈ TF , there exists r ∈ (0, 1), t > 0 such that B(x, r, t) ⊂ A. Let  = 1−r . Let y ∈ B(x, ). Then |x − y| < . Therefore F (x − y, t) > 1 − r. Indeed,

F (x − y, t) =

t t (1 − r) t t > = = 1 − r. tr = t + |x − y| t +  t + 1−r t

As F (x − y, t) > 1 − r, we obtain that y ∈ B(x, r, t). Hence B(x, ) ⊂ B(x, r, t) ⊂ A.

2

Theorem 5.8. Let X be a Hausdorff topological linear space. The following conditions are equivalent: 1. X is a fuzzy F-normed linear space; 2. X is a fuzzy metrizable topological linear space, i.e. on X we can define an invariant fuzzy metric (M(x + z, y + z, t) = M(x, y, t), (∀)x, y, z ∈ X, (∀)t > 0), which generates the topology of X; 3. X possess a countable neighbourhood base of 0;

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4. X is metrizable topological linear space; 5. The topology of X is generated by an F-norm. Proof. (1) ⇒ (2) From Theorem 3.12 and Proposition 5.3 we have that M : X × X × [0, ∞) → [0, 1] defined by M(x, y, t) = F (x − y, t) is fuzzy metric compatible with the topology of X. Moreover, as M(x + z, y + z, t) = F (x − y, t) = M(x, y, t), we have that M is invariant. (2) ⇒ (3) We suppose that the topological linear space X is fuzzy metrizable. Then, it is easy to show (see [15]), that    1 1 S(0) := B 0, , n n n∈N∗ is a countable neighbourhood base of 0. (3) ⇒ (4) For this result see [18, I, Theorem 6.1]. (4) ⇒ (5) For this result see [18, I, Theorem 6.1]. (5) ⇒ (1) Let | · | a F-norm on X. Let  t if x ∈ X, t ∈ R, t > 0 F (x, t) := t+|x| 0 if x ∈ X, t ∈ R, t ≤ 0 be the corresponding standard fuzzy F-norm on X. From the previous proposition, topology T induced by the F-norm | · | and topology TF induced by the standard fuzzy F-norm are the same. 2 Remark 5.9. In 2010, C. Alegre and S. Romaguera [1] introduced the term of fuzzy quasi-norm on a real linear space and obtain characterizations of metrizable topological linear spaces with fuzzy norms of type (N, ∗L ). Also, they proved that a paratopological linear space X is quasi-metrizable if and only the topology of X is generated by a fuzzy quasi-norm (N, ·). Definition 5.10. A complete fuzzy F-normed linear space will be called fuzzy F-space. Remark 5.11. Many examples of fuzzy F-spaces can be given. It would be interesting, though, to find an example of a fuzzy F-space with a fuzzy F-norm, which is not a fuzzy norm. Example 5.12. space of all sequences x = (xn ) of real (or complex) numbers For 0 p< p < 1, we denote by lp the linear ∞ p satisfying ∞ n=1 |xn | < ∞. The mapping d(x, y) := n=1 |xn − yn | is an invariant metric on lp and lp becomes a complete metrizable topological linear space. 1. The fuzzy set F in lp × R defined by  ∞t if x ∈ lp , t > 0 F (x, t) := t+ n=1 |xn |p 0 if x ∈ lp , t ≤ 0 is a fuzzy F-norm; 2. Topology Td induced by d and topology TF induced by the fuzzy F-norm F are the same; 3. F is not a fuzzy norm on lp . Proof. 1. It is easy to verify that F is a fuzzy F-norm on lp . We check only (F3). For x ∈ lp , t > 0, |λ| < 1, we have F (λx, t) =

t+

t ∞

p n=1 |λxn |

=

t + |λ|p

t ∞

p n=1 |xn |

If x ∈ lp , t ≤ 0, |λ| < 1, we have F (λx, t) = F (x, t) = 0.

≥

t+

t ∞

p n=1 |xn |

= F (x, t).

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2. Let A ∈ Td . We will prove that A ∈ TF . Let x ∈ A. Then there exists  > 0 such that Bd (x, ) ⊂ A. Let t = 1  1 ∈ (0, 1). We prove that BF (x, r, t) ⊂ A. Let y ∈ BF (x, r, t). Then F (x − y, t) > 1 − r = 1+ . Thus and r = 1+ ∞ 1 1 p > 1+ . Therefore n=1 |xn − yn | < . Hence y ∈ Bd (x, ) ⊂ A. Conversely, let A ∈ TF . We 1+ ∞ |x −y |p n=1

n

n

will prove that A ∈ Td . Let x ∈ A. As A ∈ TF , there exists r ∈ (0, 1), t > 0 such that BF (x, r, t) ⊂ A. Let  = p Then Bd (x, ) ⊂ A. Indeed, for y ∈ Bd (x, ), we have ∞ n=1 |xn − yn | < . Therefore F (x − y, t) =

t+

tr 1−r .

t t t (1 − r) t > = = = 1 − r. tr p t +  t + 1−r t n=1 |xn − yn |

∞

Hence y ∈ BF (x, r, t) ⊂ A. 3. Indeed, condition (N3) is not fulfilled. Let t > 0, (xn ) ∈ lp , λ ∈ K∗ . Then

 F λ(xn ), t =

t t/|λ|p = p p t + n=1 |λxn |p t + |λ|p ∞ t/|λ|p + ∞ n=1 |xn | n=1 |xn |     t t = F (xn ), p = F (xn ), . 2 |λ| |λ| t ∞

=

Remark 5.13. a) It is natural to raise the following question: does a fuzzy norm exist so that it could induce the topology of lp ? The answer is negative. In order to justify this it is enough to notice that the topology of lp is not locally convex (see [16, Section 1.47]), while the topology generated by a fuzzy norm (N, min) is one locally convex (see [1,14]). b) It must be highlighted that, if we consider fuzzy norms of (N, ·) type, the answer is positive. Thus, the fuzzy norm

p t if x ∈ lp , t > 0 p + ∞ |x |p t N(x, t) := n=1 n 0 if x ∈ lp , t ≤ 0 has been defined in paper [1]. It is easy to check that (N, ·) is a fuzzy norm and Td = TN . 6. Conclusion This paper introduces, first of all, the notion of fuzzy pseudo-norm and then it extends and improves the results obtained by T. Bag and S.K. Samanta, for fuzzy norms, in the context of fuzzy pseudo-norms. In addition, the notions of fuzzy F-norm and fuzzy F-space are introduced and discussed. We have proved that each fuzzy pseudo-norm (fuzzy F-norm) can be characterized in terms of an ascending family of pseudo-norms (F-norms). An important result of this paper is a characterization of metrizable topological linear spaces in terms of fuzzy F-norms. As an example we showed that lp (for 0 < p < 1) is a fuzzy F-space with a fuzzy F-norm, which is not a fuzzy norm. Even if the structure of fuzzy F-spaces is much more complicated than that of fuzzy Banach spaces, we have built a fertile ground to study, in further papers, fuzzy bounded linear operators on fuzzy F-spaces. We intend to establish, in this more general settings, fuzzy version of the following theorems: the Hahn–Banach theorem, the Open mapping theorem, the Closed graph theorem, the Banach–Steinhaus theorem. The resulting fuzzy F-spaces have a very rich structure, allowing, in particular, the development of the spectral theory for bounded linear operators on fuzzy F-spaces and the construction of analytic functional calculus within this fuzzy context. Acknowledgements The author is extremely grateful to the area editor and referees, for their very carefully reading of the paper and for their valuable comments and suggestions which have been useful to increase the scientific quality and presentation of the paper.

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