# A Study of Boundedness in Fuzzy Normed Linear Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**A binary operation

- 1.
- $a\ast b=b\ast a,(\forall )a,b\in [0,1]$;
- 2.
- $a\ast 1=a,(\forall )a\in [0,1]$;
- 3.
- $(a\ast b)\ast c=a\ast (b\ast c),(\forall )a,b,c\in [0,1]$;
- 4.
- If $a\le c$ and $b\le d$, with $a,b,c,d\in [0,1]$, then $a\ast b\le c\ast d$.

**Remark**

**1.**

**Definition**

**2**

**.**A t-norm ∗ is strictly monotonic if

**Remark**

**2.**

**Definition**

**3.**

**Remark**

**3.**

**Definition**

**4**

**.**The triple $(X,M,\ast )$ 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 in $X\times X\times [0,\infty )$ satisfying the following conditions:

**(M1)**- $M(x,y,0)=0,(\forall )x,y\in X$;
**(M2)**- $(\forall )x,y\in X,x=y$ if and only if $M(x,y,t)=1$ for all $t>0$;
**(M3)**- $M(x,y,t)=M(y,x,t),(\forall )x,y\in X,(\forall )t>0$;
**(M4)**- $M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s),(\forall )x,y,z\in X,(\forall )t,s>0$;
**(M5)**- $(\forall )x,y\in X,M(x,y,\xb7):[0,\infty )\to [0,1]$ is left continuous and $\underset{t\to \infty}{lim}M(x,y,t)=1$.

**Definition**

**5**

**.**Let $(X,M,\ast )$ be a fuzzy metric space. A subset A of X is said to be F-bounded if

**Definition**

**6**

**.**Let X be a vector space over a field $\mathbb{K}$ (where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$) and ∗ be a continuous t-norm. A fuzzy set N in $X\times [0,\infty )$ is called a fuzzy norm on X if it satisfies:

**(N1)**- $N(x,0)=0,(\forall )x\in X$;
**(N2)**- $[N(x,t)=1,(\forall )t>0]$ if and only if $x=0$;
**(N3)**- $N(\lambda x,t)=N\left(x,\frac{t}{\left|\lambda \right|}\right),(\forall )x\in X,(\forall )t\ge 0,(\forall )\lambda \in {\mathbb{K}}^{\ast}$;
**(N4)**- $N(x+y,t+s)\ge N(x,t)\ast N(y,s),(\forall )x,y\in X,(\forall )t,s\ge 0$;
**(N5)**- $(\forall )x\in X$, $N(x,\xb7)$ is left continuous and $\underset{t\to \infty}{lim}N(x,t)=1$.

**Example**

**1**

**.**Let $(X,||\xb7||)$ be a normed linear space. Let $N:X\times [0,\infty )\to [0,1]$ defined by

**Theorem**

**1**

- 1.
- We define $M:X\times X\times [0,\infty )\to [0,1]$ by $M(x,y,t)=N(x-y,t)$. Then M is a fuzzy metric on X.
- 2.
- For $x\in X,\alpha \in (0,1),t>0$ we define the open ball$$B(x,\alpha ,t):=\{y\in X\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}N(x-y,t)>1-\alpha \}\phantom{\rule{0.277778em}{0ex}}.$$Then$${\mathcal{T}}_{N}:=\left\{T\subset X\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}x\in T\phantom{\rule{0.277778em}{0ex}}iff\phantom{\rule{0.277778em}{0ex}}(\exists )t>0,(\exists )\alpha \in (0,1)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}B(x,\alpha ,t)\subseteq T\right\}$$

**Theorem**

**2.**

**Proof.**

**Definition**

**7**

**.**Let $(X,N,\ast )$ be a FNL space and $({x}_{n})$ be a sequence in X. The sequence $({x}_{n})$ is said to be convergent if $(\exists )x\in X$ such that $\underset{n\to \infty}{lim}N({x}_{n}-x,t)=1\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}(\forall )t>0\phantom{\rule{0.277778em}{0ex}}.$ In this case, x is called the limit of the sequence $({x}_{n})$ and we denote $\underset{n\to \infty}{lim}{x}_{n}=x$ or ${x}_{n}\to x$.

**Definition**

**8**

**.**Let $(X,N,\ast )$ be a FNL space. A subset B of X is called the closure of the subset A of X if for any $x\in B$, $(\exists )({x}_{n})\subset A$ such that ${x}_{n}\to x$. We denote the set B by $\overline{A}$.

**Remark**

**4.**

**Definition**

**9**

**.**A subset A of a FNL space X is said to be bounded if

**Definition**

**10**

**.**A subset A of a FNL space X is called fuzzy bounded if

**Definition**

**11**

**.**A subset A of a FNL space X is called fuzzy totally bounded if

## 3. Fuzzy Bounded Sets

**Theorem**

**3.**

**Proof.**

**Remark**

**5.**

**Theorem**

**4.**

**Proof.**

**Remark**

**6.**

**Corollary**

**1**

- 1.
- If $A,B$ are fuzzy bounded, then $A\cup B$ and $A+B$ are fuzzy bounded;
- 2.
- If A is fuzzy bounded, then $\overline{A}$ is fuzzy bounded.

**Corollary**

**2.**

**Proposition**

**1.**

**Proof.**

## 4. Bounded Sets

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 5. F-Bounded Sets

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

## 6. Fuzzy Totally Bounded Sets

**Theorem**

**5.**

- 1.
- A is fuzzy totally bounded;
- 2.
- $(\forall )\alpha \in (0,1),(\exists )\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}\subset A\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}A\subset {\displaystyle \bigcup _{i=1}^{n}}({x}_{i}+B(0,\alpha ,\alpha ))$;
- 3.
- $(\forall )\alpha \in (0,1),(\forall )t>0,(\exists )\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}\subset A\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}A\subset {\displaystyle \bigcup _{i=1}^{n}}({x}_{i}+B(0,\alpha ,t))$;
- 4.
- $(\forall )\alpha \in (0,1),(\forall )t>0,(\exists )\{{x}_{1},{x}_{2},\cdots ,{x}_{n}\}\subset X\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}A\subset {\displaystyle \bigcup _{i=1}^{n}}({x}_{i}+B(0,\alpha ,t))$.

**Proof.**

**Proposition**

**8.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

- 1.
- A set M is bounded in $(X,||\xb7||)$ if and only if M is fuzzy bounded in $(X,N,\ast )$;
- 2.
- A set M is totally bounded in $(X,||\xb7||)$ if and only if M is fuzzy totally bounded in $(X,N,\ast )$.

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 7. A Comparative Study among Different Types of Boundedness

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Theorem**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

## 8. Conclusions and Further Works

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Bînzar, T.; Pater, F.; Nădăban, S.
A Study of Boundedness in Fuzzy Normed Linear Spaces. *Symmetry* **2019**, *11*, 923.
https://doi.org/10.3390/sym11070923

**AMA Style**

Bînzar T, Pater F, Nădăban S.
A Study of Boundedness in Fuzzy Normed Linear Spaces. *Symmetry*. 2019; 11(7):923.
https://doi.org/10.3390/sym11070923

**Chicago/Turabian Style**

Bînzar, Tudor, Flavius Pater, and Sorin Nădăban.
2019. "A Study of Boundedness in Fuzzy Normed Linear Spaces" *Symmetry* 11, no. 7: 923.
https://doi.org/10.3390/sym11070923