**2. Preliminaries**

In this section, we recall some basic notion, definitions and lemma that are required for the following sections.

*Definition* 2.1. A mapping *F* : ↦ *λ*ð Þ ¼ ½ � 0, 1 is a real fuzzy number on the set associating real number *s* with its grade of membership *F s*ð Þ. Let C denote the set of all closed and bounded intervals *F* ¼ *f* <sup>1</sup>, *f* <sup>2</sup> on the real line . For *<sup>G</sup>* <sup>¼</sup> *<sup>g</sup>*1, *<sup>g</sup>*<sup>2</sup> in <sup>C</sup>, one define *F* ≤ *G* if and only if *f* <sup>1</sup> ≤ *f* <sup>2</sup> and *g*<sup>1</sup> ≤ *g*2. Determine a metric *ρ* on C by

$$\rho(F, G) = \max\left\{|f\_1 - \mathbf{g}\_1|, |f\_2 - \mathbf{g}\_2|\right\}.\tag{1}$$

It can be easily seen that "≤ " is a partial order on C and ð Þ C, *ρ* is a complete metric space. The absolute value ∣*F*∣ of *F* ∈ ð Þ*λ* is defined by

$$|F|(s) = \begin{cases} \max\left\{ F(s), F(-s) \right\}, & \text{if } s > 0\\ 0, & \text{if } |s < 0. \end{cases}$$

Suppose *ρ* : ð Þ� *λ* ð Þ*λ* ↦ be determined as

$$\overline{\rho}(F,G) = \sup \rho(F,G).$$

Hence, *ρ* defines a metric on ð Þ*λ* . The multiplicative and additive identity in ð Þ*λ* are denoted by 1 and 0, respectively.

*Definition* 2.2. A family of subsets J of the power set *P*ð Þ of the natural number is known as an ideal if and only if the following conditions are satisfied [8]

i. ∅ ∈J ,

ii. for every A1, A<sup>2</sup> ∈J one obtain A<sup>1</sup> ∪ A<sup>2</sup> ∈J ,

iii. for every A<sup>1</sup> ∈J and every A<sup>2</sup> ⊆ A<sup>1</sup> one obtain A<sup>2</sup> ∈J .

An ideal J is known as non– trivial if J 6¼ *P*ð Þ and non– trivial ideal is said to be an admissible if J ⊇ f g f g*n* : *n*∈ .

**Lemma 1.** *If ideal* J *is maximal, then for every* A ⊂ *we have either* A ∈J *or* nA ∈J [8]*.*

*Definition* 2.3. A family of subsets H of the power set *P*ð Þ of the natural number is known as filter in if and only if following condition are satisfied [8].

i. ∅ ∉ H,


*Remark* 1. Filter associated with the ideal J is defined by the family of sets

$$\mathcal{H}(\mathcal{I}) = \{ K \subset \mathbb{N} : \exists \quad \mathcal{A} \in \mathcal{I} : K = \mathbb{N} \backslash \mathcal{A} \}$$

*Definition* 2.4. A sequence ð Þ *Fk* of fuzzy real numbers is known as J � convergent to fuzzy real numbers *F*<sup>0</sup> if for each *ε*>0, the set [9].

$$\{k \in \mathbb{N} : \overline{\rho}(F\_k, F\_0) \ge \varepsilon\} \in \mathcal{I}.\,\,\,$$

*Definition* 2.5. A sequence ð Þ *Fk* is known as J � null if there exists a fuzzy real numbers 0 such that for each *ε*>0 [9],

$$\left\{ k \in \mathbb{N} : \overline{\rho}(F\_k, \overline{\mathbf{0}}) \ge \varepsilon \right\} \in \mathcal{I}.$$

*Definition* 2.6. A sequence ð Þ *Fk* of fuzzy real numbers is known as J -Cauchy if there exists a subsequence *Fl<sup>ε</sup>* ð Þ of ð Þ *Fk* in such a way that for every *ε*>0 [13],

$$\left\{ k \in \mathbb{N} : \overline{\rho}(F\_k, F\_{l\_\*}) \ge \varepsilon \right\} \in \mathcal{I}.$$

*Definition* 2.7. A sequence ð Þ *Fk* is known as J � bounded if there exists a fuzzy real numbers M>0 so that, the set [9].

$$\left\{ k \in \mathbb{N} : \overline{\rho}(F\_k, \overline{0}) > \mathcal{M} \right\} \in \mathcal{I}.$$

*Definition* 2.8. Suppose *K* ¼ f g *ki* ∈ : *k*<sup>1</sup> <*k*<sup>2</sup> < … ⊆ and be a sequence space. A *K*– step space of *E* is a sequence space [12].

$$\Lambda\_K^{\mathbb{E}} = \left\{ (\mathfrak{x}\_{k\_i}) \in o \, : \, (\mathfrak{x}\_k) \in \mathbb{E} \right\}.$$

The canonical pre-image of a sequence *xki* ð Þ<sup>∈</sup> *<sup>λ</sup> <sup>K</sup>* is a sequence *yk* � �∈*ω* defined as follows:

$$\mathcal{y}\_k = \begin{cases} \mathfrak{x}\_k, & \text{if} \quad n \in K \\ \mathbf{0}, & \text{otherwise.} \end{cases}$$

*y* is in canonical pre-image of Λ *<sup>K</sup>* if *y* is canonical pre-image of some element *x*∈Λ *K:*

*Definition* 2.9. A sequence space is known as monotone, if it is contains the canonical pre-images of it is step space [12].

That is, if for all infinite *K* ⊆ and ð Þ *xk* ∈ the sequence ð Þ *αkxk* , where *α<sup>k</sup>* ¼ 1 for *k*∈*K* and *α<sup>k</sup>* ¼ 0 otherwise, belongs to .

*Definition* 2.10. [12] A sequence space is known as convergent free, if ð Þ *xk* ∈ whenever *yk* � �∈ and *yk* � � <sup>¼</sup> 0 implies that ð Þ¼ *xk* 0 for all *<sup>k</sup>*<sup>∈</sup> .

**Lemma 2.1.** Every solid space is monotone [12].

*Definition* 2.11. Suppose *U* and *V* are normed spaces. An operator T : *U* ! *V* is known as compact linear operator if [12].

1.T is linear

2.T maps every bounded sequence ð Þ *xk* in *U* onto a sequence T ð Þ *xk* in *V* which has a convergent subsequence.

#### **3. Main results**

In this section, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate some topological and algebraic properties on these spaces. We denote *ω<sup>F</sup>* the class of all sequences of fuzzy real numbers and J be an admissible ideal of the subset of the natural numbers .

$${}\_{F}\mathcal{S}^{\mathcal{I}\_{0}}(T) = \left\{ F = (F\_{k}) \in \boldsymbol{o}\boldsymbol{o}^{F} : \{ \boldsymbol{r} \in \mathbb{N} : \boldsymbol{h}\_{r}^{-1} \sum\_{k \in \mathcal{I}\_{r}} \overline{\rho}(\boldsymbol{T}(F\_{k}), \boldsymbol{F}\_{0}) \ge \boldsymbol{e} \} \in \mathcal{J} \quad \text{for some} \quad {}^{F}\boldsymbol{e} \in \mathbb{R}(\boldsymbol{\mathcal{I}}) \right\},$$

$${}\_{F}\mathcal{S}^{\mathcal{I}\_{0}}\_{0}(T) = \left\{ F = (F\_{k}) \in \boldsymbol{o}^{F} : \{ \boldsymbol{r} \in \mathbb{N} : \boldsymbol{h}\_{r}^{-1} \sum\_{k \in \mathcal{I}\_{r}} \overline{\rho}(\boldsymbol{T}(F\_{k}), \overline{\mathbf{0}}) \ge \boldsymbol{e} \} \in \mathcal{J} \right\},$$

$${}\_{F}\mathcal{S}^{\mathcal{I}\_{\infty}}\_{\infty}(T) = \left\{ F = (F\_{k}) \in \boldsymbol{o}^{F} : \exists \quad \mathcal{M} > 0 : \{ \boldsymbol{r} \in \mathbb{N} : \boldsymbol{h}\_{r}^{-1} \sum\_{k \in \mathcal{I}\_{r}} \overline{\rho}(\boldsymbol{T}(F\_{k}), \overline{\mathbf{0}}) \ge \boldsymbol{\mathcal{M}} \} \in \mathcal{J} \right\},$$

$${}\_{F}\mathcal{S}^{\emptyset}\_{\infty}(T) = \left\{ F = (F\_{k}) \in \boldsymbol{o}^{F} : \sup\_{\boldsymbol{r}} \boldsymbol{h}\_{r}^{-1} \sum\_{k \in \mathcal{I}\_{r}} \overline{\rho}(\boldsymbol{T}(F\_{k}), \overline{\mathbf{0}}) < \infty \right\}.$$

**Theorem 3.1.** The sequence spaces *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T , *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T and *<sup>F</sup>*S<sup>J</sup> <sup>∞</sup>ð Þ T are linear spaces. *Proof.* Suppose *α* and *β* be scalars, and assume that *F* ¼ ð Þ *Fk* , *G* ¼ *gk* � �∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> .

Since *Fk*, *Gk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Then for a given *ε*>0, there exists *F*1, *F*<sup>2</sup> ∈ℂ in such a manner that

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in f\_r} \overline{\rho}(T(F\_k), F\_1) \ge \frac{\varepsilon}{2} \right\} \in \mathcal{I} \,.$$

and

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F\_k), F\_2) \ge \frac{\varepsilon}{2} \right\} \in \mathcal{I}.$$

Now, let

$$A\_1 = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F\_k), F\_1) < \frac{t}{2|\alpha|} \right\} \in \mathcal{H}(\mathcal{J}).$$

$$A\_2 = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F\_k), F\_2) < \frac{t}{2|\beta|} \right\} \in \mathcal{H}(\mathcal{J}).$$

be such that *A<sup>c</sup>* 1, *A<sup>c</sup>* <sup>2</sup> ∈J . Therefore, the set

$$\begin{split} A\_3 &= \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (T(\alpha(F\_k) + \beta(\mathcal{G}\_k)), (aL\_1 + \beta L\_2)) < \varepsilon \right\} \\ &\supseteq \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (T(F\_k), L\_1) < \frac{\varepsilon}{2|a|} \right\} \cap \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (T(F\_k), L\_1) < \frac{\varepsilon}{2|\beta|} \right\}. \end{split} \tag{2}$$

Thus, the sets on right hand side of (2) belong to H Jð Þ. Therefore *Ac* <sup>3</sup> belongs to J . Therefore, *α*ð Þþ *Fk β*ð Þ *Gk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Hence *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T is linear space.

In similar manner, one can easily prove that *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T and *<sup>F</sup>*S<sup>J</sup> <sup>∞</sup>ð Þ T are linear spaces. □

**Example 3.1.** Suppose J ¼J *<sup>ρ</sup>* ¼ Bf g ⊆ : *ρ*ð Þ¼ B 0 , where *ρ*ð Þ B denotes the asymptotic density of B. In this case *<sup>F</sup>*S<sup>J</sup> *<sup>ρ</sup>* ð Þ¼ T *<sup>F</sup>*S*θ*ð Þ T , where

*A Study of Fuzzy Sequence Spaces DOI: http://dx.doi.org/10.5772/intechopen.94202*

$$\mathcal{S}\_F \mathcal{S}\_\theta(T) = \left\{ F\_k \in \alpha^F : \rho\left( \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(T(F\_k), F\_0) \ge \varepsilon \right\} \right) = 0 \quad \text{for some} \quad F\_0 \in \mathbb{R}(\chi) \right\}.$$

**Example 3.2.** Suppose J ¼J *<sup>f</sup>* ¼ Bf g ⊆ : B is finite . J *<sup>f</sup>* is an admissible ideal in and *<sup>F</sup>*S<sup>J</sup> *<sup>f</sup>*ð Þ¼ <sup>T</sup> *<sup>F</sup>*S*<sup>θ</sup>* ð Þ T .

**Theorem 3.2.** The spaces *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ *T* and *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ *T* are not convergent free. *Proof.* For the proof of the theorem, we consider the subsequent example. □ **Example 3.3.** Suppose J ¼J *<sup>δ</sup>* and T ð Þ¼ *Fk Fk*. Consider the sequence *Fk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T as: For *k* 6¼ *i* 2 , *i*∈

$$F\_k(s) = \begin{cases} \mathbf{1}, & \text{if} \quad \mathbf{0} \le s \le k^{-1} \\\\ \mathbf{0}, & \text{otherwise}. \end{cases}$$

For *k* ¼ *i* 2 , *i*∈ , *Fk*ðÞ¼ *s* 0. Therefore

$$F\_k = \begin{cases} [\mathbf{0}, \mathbf{0}], & \text{if } \ k^2 = i \\\\ [\mathbf{0}, k^{-1}], & k \neq i^2. \end{cases}$$

Hence, *Fk* ! 0 *as k* ! ∞. Thus *Fk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Let *Gk* be sequence in such a way that, for *k* ¼ *i* 2 , *i* ∈ , *Fk*ðÞ¼ *s* 0. Therefore, one obtain

$$\mathbf{G}\_k(s) = \begin{cases} \mathbf{1}, & \text{if} \quad \mathbf{0} \le s \le k, \\\\ \mathbf{0}, & \text{otherwise.} \end{cases}$$

For *k* ¼ *i* 2 , *i*∈ , *Fk*ðÞ¼ *s* 0. Therefore

$$G\_k = \begin{cases} [\mathbf{0}, \mathbf{0}], & \text{if} \quad k^2 = i \\\\ [\mathbf{0}, k], & k \neq i^2. \end{cases}$$

It can be easily seen that *Gk* ∉ *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T .

Hence *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T and *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T are not convergence free.

**Theorem 3.3.** The sequence *<sup>F</sup>* <sup>¼</sup> ð Þ *Fk* <sup>∈</sup>*F*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T is J � convergent if and only if for every *ε* >0, there exists *N<sup>ε</sup>* ∈ in such a way that

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k), \mathcal{T}(F\_{N\_\varepsilon})) < \varepsilon \right\} \in \mathcal{H}(\mathcal{I}).\tag{3}$$

*Proof.* Suppose that *<sup>F</sup>* <sup>¼</sup> ð Þ *Fk* <sup>∈</sup> *<sup>F</sup>*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T Let *F*<sup>0</sup> ¼J� lim *F*. Then for every *ε*>0, the set

$$B\_{\varepsilon} = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k), F\_0) < \frac{\varepsilon}{2} \right\} \in \mathcal{H}(\mathcal{I}).$$

Fix an *N<sup>ε</sup>* ∈*Bε*. Then, we have

*Fuzzy Systems - Theory and Applications*

$$
\overline{\rho}(\mathcal{T}(F\_k), \mathcal{T}(F\_{N\_\varepsilon})) \le \overline{\rho}(\mathcal{T}(F\_k), F\_0)
$$

$$
+ \overline{\rho}(\mathcal{T}(F\_{N\_\varepsilon}), F\_0) < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.
$$

Which holds for all *N<sup>ε</sup>* ∈*Bε*. Hence

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k), \mathcal{T}(F\_{N\_\epsilon})) < \varepsilon \right\} \in \mathcal{H}(\mathcal{I}).$$

On Contrary, assume that

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k), \mathcal{T}(F\_{N\_\epsilon})) < \varepsilon \right\} \in \mathcal{H}(\mathcal{I}).$$

That is

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(\mathcal{T}(F\_k), \mathcal{T}(F\_{N\_r})) < \varepsilon \right\} \in \mathcal{H}(\mathcal{I}) \text{ for all } \varepsilon > 0. \right\}$$

Then, the set

$$\mathcal{C}\_{\varepsilon} = \{ \mathbb{k} \in \mathbb{N} : \mathcal{T}(F\_k) \in [\mathcal{T}(F\_{N\_{\varepsilon}}) - \varepsilon, \mathcal{T}(F\_{N\_{\varepsilon}}) + \varepsilon] \} \in \mathcal{H}(\mathcal{J}) \text{ for all } \varepsilon > 0.$$

Let *I<sup>ε</sup>* ¼ T *FN<sup>ε</sup>* ð Þ� *ε*, T *FN<sup>ε</sup>* ½ � ð Þþ *ε* . If we fix an *ε*>0, then we have *C<sup>ε</sup>* ∈ H Jð Þ as well as *C<sup>ε</sup>* <sup>2</sup> ∈ H Jð Þ. Hence *C<sup>ε</sup>* <sup>2</sup> ∩*C<sup>ε</sup>* ∈ H Jð Þ. This implies that *I* ¼ *I<sup>ε</sup>* ∩ *I<sup>ε</sup>* <sup>2</sup> 6¼ ∅. That is

$$\{k \in \mathbb{N} : \mathcal{T}(F\_k) \in I\} \in \mathcal{H}(\mathcal{I}).$$

That is

$$\text{diam} \quad I \le \text{diam} \quad I\_v.$$

Where the diam of *I* denotes the length of interval *I*. Continuing in this way, by induction, we get the sequence of closed intervals.

$$I\_{\iota} = \mathcal{I}\_0 \supseteq \mathcal{I}\_1 \supseteq \dots \supseteq \mathcal{I}\_k \supseteq \dots$$

With the property that

$$\text{diam} \quad \mathcal{I}\_k \le \frac{1}{2} \quad \text{diam} \quad \mathcal{I}\_{k-1} \text{ for } \left(k = 2, 3, 4, \dots\right).$$

and

$$\{k \in \mathbb{N} : \mathcal{T}(F\_k) \in \mathcal{T}\_k\} \in \mathcal{H}(\mathcal{I}) \text{ for } (k = 2, 3, 4, \dots).$$

Then there exists a *ξ*∈ ∩ I*<sup>k</sup>* where *k*∈ in such a way that *ξ* ¼J� lim *T F*ð Þ*<sup>k</sup>* . Therefore, the result holds. □

**Theorem 3.4.** *The inclusions <sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T ⊂ *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>∞</sup> ð Þ T *hold.* *Proof.* Let *F* ¼ ð Þ *Fk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Then there exists a number *F*<sup>0</sup> ∈ such that

$$\mathcal{J} - \lim\_{k \to \infty} \overline{\rho}(\mathcal{T}(F\_k), F\_0) = \mathbf{0}.$$

That is, the set

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F\_k), F\_0) \ge \varepsilon \right\} \in \mathcal{I} \,.$$

Here

$$\begin{aligned} h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} \left( \mathcal{T}(F\_k), \overline{\mathbf{0}} \right) &= h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (\mathcal{T}(F\_k), -F\_0 + F\_0) \\ &\le h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (\mathcal{T}(F\_k) - F\_0) + h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (\mathcal{T}(F\_k), F\_0) \end{aligned} \tag{4}$$

On the both sides, taking the supremum over *k* of the above equation, we obtain *Fk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Therefore inclusion holds.

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho} (\mathcal{T}(F\_k^p), a\_p) \ge \varepsilon \right\} \in \mathcal{I} \text{ .}$$

Then, it proves that *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Hence *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ T ⊂*F*S<sup>J</sup> *<sup>θ</sup>* <sup>∞</sup> ð Þ T . □

**Theorem 3.5.** The space *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ *T* is neither normal nor monotone if J is not maximal ideal.

**Example 3.4.** Suppose fuzzy number

$$G\_k(s) = \begin{cases} \frac{1+s}{2}, & \text{if } \quad -1 \le s \le 1\\ \frac{3-s}{2}, & \text{if } \quad 1 \le s \le 3\\ 0, & \text{otherwise.} \end{cases}$$

Then *Gk*ð Þ*s* ∈ *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Applying Lemma 1, there exists a subset *K* of in such a way that *K* ∉ J and n*K* ∉ J . Determine a sequence *G* ¼ ð Þ *Gk* as

$$G\_k = \begin{cases} F\_k, & k \in K \\ \mathbf{0}, & \text{otherwise.} \end{cases}$$

Therefore ð Þ *Gk* belong to the canonical pre- image of the *K*- step spaces of *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . But ð Þ *Gk* ∉ *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T . Hence *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T is not monotone. Hence, by Lemma (2.1) *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T is not normal.

**Theorem 3.6.** The sequence space *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T is solid and monotone. *Proof.* Suppose *F* ¼ ð Þ *Fk* ∈ *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T , then for *ε*> 0, the set

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in f\_r} \overline{\rho}(\mathcal{T}(F\_k), F\_0) \ge \varepsilon \right\} \in \mathcal{I}. \tag{5}$$

Suppose that sequence of scalars ð Þ *α<sup>k</sup>* with the property ∣*αk*∣ ≤1 ∀ *k*∈ . Therefore

$$\begin{aligned} \overline{\rho}(\mathcal{T}(a\_k F\_k), F\_0) &= \overline{\rho}(a\_k \mathcal{T}(F\_k), F\_0) \\ \leq & |a\_k| \overline{\rho}(\mathcal{T}(F\_k), F\_0) \quad \text{for all} \ k \in \mathbb{N}. \end{aligned} \tag{6}$$

Hence, from the Eq. (5) and above inequality, one obtain

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho} (\mathcal{T}(a\_k F\_k), F\_0) \ge \varepsilon \right\}$$

$$\subseteq \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho} (\mathcal{T}(F\_k), F\_0) \ge \varepsilon \right\} \in \mathcal{T}.$$

Implies

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(a\_k F\_k), F\_0) \ge \varepsilon \right\} \in \mathcal{I} \text{ .}$$

Therefore, ð Þ *αkFk* ∈*F*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T . Then *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T is solid and monotone by lemma 2.1. □

**Theorem 3.7.** The sequence space *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> is closed subspace of *<sup>F</sup>*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T . *Proof.* Suppose *F<sup>q</sup> k* � � be a Cauchy sequence in *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> . Then, *<sup>F</sup><sup>q</sup> <sup>k</sup>* ! *F* in *<sup>F</sup>*S∞ð Þ T as *<sup>q</sup>* ! <sup>∞</sup>. Since *<sup>F</sup><sup>q</sup> k* � �∈*F*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> , then for each *<sup>ε</sup>*<sup>&</sup>gt; 0 there exists *aq* such that converges to *a*.

$$\mathcal{J} - \operatorname{lim} F = a.$$

Since *F<sup>q</sup> k* � � be a Cauchy sequence in *<sup>F</sup>*S*<sup>I</sup><sup>θ</sup>* ð Þ <sup>T</sup> . Then for each *<sup>ε</sup>*>0 there exists *n*<sup>0</sup> ∈ , such that

$$\sup\_{k} \overline{\rho} \left( \mathcal{T} \left( F\_k^q \right), \mathcal{T} \left( F\_k^s \right) \right) < \frac{\varepsilon}{3} \quad \text{for all} \ q, s \ge n\_0. 1$$

For a given *ε*>0,

$$B\_{q,s} = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho} (\mathcal{T} \left( F\_k^q \right), \mathcal{T} \left( F\_k^s \right)) < \frac{\varepsilon}{3} \right\}.$$

Now,

$$B\_q = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k^{\overline{p}}), a\_q) < \frac{\varepsilon}{3} \right\}.$$

and

$$B\_{\varepsilon} = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho} (\mathcal{T} \left( F\_k^{\varepsilon} \right), a\_s) < \frac{\varepsilon}{3} \right\}.$$

**36**

*A Study of Fuzzy Sequence Spaces DOI: http://dx.doi.org/10.5772/intechopen.94202*

Then *Bc q*,*s* , *Bc <sup>q</sup>*, *Bc <sup>s</sup>* <sup>∈</sup><sup>J</sup> . Let *Bc* <sup>¼</sup> *Bc <sup>q</sup>*,*<sup>s</sup>* ∪ *Bc <sup>q</sup>* ∪ *Bc <sup>s</sup>*. Where

$$B = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(a\_q, a\_s) < \epsilon \right\} \text{ then } \ B^\epsilon \in \mathcal{I}.$$

Assume *n*<sup>0</sup> ∈*Bc* . Then, for every *q*, *s*≥ *n*<sup>0</sup> it has

$$\begin{aligned} \overline{\rho}\left(a\_q, a\_s\right) &\leq \overline{\rho}\left(\mathcal{T}\left(F\_k^q\right), a\_q\right) + \overline{\rho}\left(\mathcal{T}\left(F\_k^s\right), a\_s\right) + \overline{\rho}\left(\mathcal{T}\left(F\_k^q\right), \mathcal{T}\left(F\_k^s\right)\right) \\ &< \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon. \end{aligned}$$

Hence, *aq* is a Cauchy sequence of scalars in ℂ, so there exists scalar *a*∈ℂ in such a way that *aq* � � ! *<sup>a</sup>* as *<sup>q</sup>* ! <sup>∞</sup>. For this step, let 0<*<sup>α</sup>* <sup>&</sup>lt;1 be given. Therefore it proved that whenever

$$U = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F), a) < a \right\} \text{ then } \ U^{\epsilon} \in \mathcal{J}. \tag{7}$$

Since *F<sup>q</sup> k* � � ! *<sup>F</sup>*, there exists *<sup>q</sup>*<sup>0</sup> <sup>∈</sup> so that

$$P = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(T(F^{q\_0}), T(F)) < \frac{a}{3} \right\}.$$

implies that *Pc* <sup>∈</sup><sup>J</sup> . The number, *qo* can be chosen together with Eq. (7), it have

$$Q = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in f\_r} \overline{\rho}(a\_{q\_0}, a) < \frac{a}{3} \right\}.$$

Which implies that *<sup>Q</sup><sup>c</sup>* <sup>∈</sup><sup>J</sup> . Since

$$\left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho} (\mathcal{T} \left( F\_k^{q\_0} \right), a\_{q\_0}) \ge \frac{a}{3} \right\} \in \mathcal{I} \,.$$

Then it has a subset *<sup>S</sup>* such that *<sup>S</sup><sup>c</sup>* <sup>∈</sup><sup>J</sup> , where

$$S = \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in J\_r} \overline{\rho}(\mathcal{T}(F\_k^{q\_0}), a\_{q\_0}) < \frac{a}{3} \right\}.$$

Suppose that *<sup>U</sup><sup>c</sup>* <sup>¼</sup> *<sup>P</sup><sup>c</sup>* <sup>∪</sup> *<sup>Q</sup><sup>c</sup>* <sup>∪</sup> *<sup>S</sup><sup>c</sup>* . Then, for every *k*∈ *U<sup>c</sup>* it has

$$\begin{cases} r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(\mathcal{T}(F), a) < a \right\} \supseteq \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(\mathcal{T}(F^{q\_0}), (F)) < \frac{a}{3} \right\} \cap \\\\ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(a\_{q\_0}, a) < \frac{a}{3} \right\} \cap \left\{ r \in \mathbb{N} : h\_r^{-1} \sum\_{k \in I\_r} \overline{\rho}(\mathcal{T}(F\_k^{q\_0}), a\_{q\_0}) < \frac{a}{3} \right\}. \end{cases} \tag{8}$$

The right hand side of Eq. (8) belongs to H Jð Þ. Hence, the sets on the left hand side of Eq. (8) belong to H Jð Þ. Therefore its complement belongs to J . Thus, J � lim *<sup>ρ</sup> aq*<sup>0</sup> , *<sup>a</sup>* � � <sup>¼</sup> 0. □

In the following example to prove that *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> is closed subspace of *<sup>F</sup>*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T . **Example 3.5.** Suppose that sequence of fuzzy number determine by

$$F\_k(s) = \begin{cases} 2^{-1}(1+s), & \text{for } s \in [-1, 1] \\ 2^{-1}(3-s), & \text{for } s \in [1, 3] \\ 0, & \text{otherwise.} \end{cases}$$

Hence, *<sup>ρ</sup> Fk*, <sup>0</sup> � � <sup>¼</sup> sup*<sup>ρ</sup> Fk*, <sup>0</sup> � �. Therefore ð Þ *Fk* <sup>∈</sup> *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> and *<sup>L</sup>* <sup>¼</sup> 0. So, it can be easily seen that *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> is closed subspace of *<sup>F</sup>*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T .

### **4. Conclusion**

The spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate algebraic and topological properties together with some examples on the given spaces. We proved that the new introduced sequence spaces are linear. Some spaces are convergent free and we also proved that space *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ <sup>T</sup> is closed subspace of *<sup>F</sup>*S*<sup>θ</sup>* <sup>∞</sup>ð Þ T . These new spaces and results provide new tools to help the authors for further research and to solve the engineering problems.
