Diminnie, R. Freese [1] and many others developed new concept like **linear** **2**-**normed** space.The notion of an **almost** **linear** **space** (als) was introduced by G. Godini [**2**]-[6]. All spaces involved in this work are over the real field ℝ.S. Gahler, Y.J. Cho, C. S. Elumalai, R. Vijayaragavan [12]-[13] established some characterization of best approximation in terms of **2**-semi inner products and normalized duality mapping associated with a **linear** **2**-**normed** **space**. Basing on this we introduced a new concept called2-**normed** **almost** **linear** spaceand established some results of best approximation in **2**- **normed** **almost** **linear** **space**[17] andsome results of best simultaneous approximation in **2**- **normed** **almost** **linear** **space** [18] . Mehmet A¸cıkg¨oz [16] introduced the concept𝜀-Approximation in generalized **2**- **normed** **linear** **space** and established some results. Basing on this we introduced a new concept calledδ- Best approximation in **2**-**normed** **almost** **linear** **space** in this paper and we established some results on δ- best approximation and δ- orthogonalisation in **2**- **normed** **almost** **linear** **space**.

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In 1965, the theory of fuzzy sets was introduced by L. Zadeh [9]. In 1964, a satisfactory theory of **2**-norm on a **linear** **space** has been introduced and developed by Gahler [**2**]. In 2003, the concepts fuzzy norm and α-norm were introduced by Bag and Samanta [1]. Jialu Zhang [3] has defined fuzzy **linear** **space** in a different way. The notion of **2**-fuzzy **2**-**normed** **linear** **space** of the set of all fuzzy sets of a set was introduced by R.M. Somasundaram and Thangaraj Beaula [6]. The concept of intuitionistic 2fuzzy **2**-**normed** **linear** **space** of the set of all fuzzy sets of a set was introduced by Thangaraj Beaula and Lilly Esthar Rani [7].

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The concepts involving continuity play a major role not only in pure mathematics but also in other branches of sciences like computer science, information theory etc., Menger [6] introduced the notion called a generalized metric in 1928. On the other hand Vulich [11] in 1938 defined a notion of higher dimensional norm in **linear** spaces. The concept of **2**-**normed** **space** was developed by Gahler in the middle of 1960’s [5]. Recently many mathematicians came out with results in **2**-**normed** spaces, analogous with that in classical **normed** spaces and Banach spaces [4,7,8]. Using the main idea in the definition of sequential continuity, many kinds of continuities were introduced and investigated in [1,2,3,10] not all but only some of them. The concept of ward continuity of real functions and ward compactness of a subset E of R are introduced by Cakalli in [**2**]. In 1965, Zadeh [12] introduced the concept of the fuzzy set in this seminal paper. Somasundaram and Beaula [9] have newly coined **2**-fuzzy **normed** **linear** **space** and proved many important theorems. In this paper our aim is to investigate the concept of fuzzy ward continuity in an intuitionistic **2**-fuzzy **2**-**normed** **linear** **space** and prove some interesting theorems. **2**. Preliminaries

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Recently, Somasundaram and Beaula [10] introduced the concept of **2**-fuzzy **2**- **normed** **linear** **space** or fuzzy **2**-**normed** **linear** **space** of the set of all fuzzy sets of a set. The authors gave the notion of a-**2**-norm on a **linear** **space** corresponding to the **2**- fuzzy **2**-norm by using some ideas of Bag and Samanta [8] and also gave some funda- mental properties of this **space**.

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The concept of fuzzy set was introduced by Zadeh [8] in 1965. A satisfactory theory of **2**-norm on a **linear** **space** has been introduced and developed by Gahler in [3]. The concept of fuzzy norm and α-norm were introduced by Bag and Samanta and the notions of convergence and Cauchy sequence were also discussed in [1]. Jialuzhang [4] has defined fuzzy **linear** **space** in a different way. Bag and Samanta [**2**] defined fuzzy bounded **linear** operator and fuzzy bounded **linear** functional on the fuzzy dual **space**. Somasundaram and Beaula [6] defined the notion of **2**-fuzzy **2**-**normed** **linear** **space**. They also established the famous closed graph theorem and Riesz theorem in **2**-fuzzy **2**-**normed** **linear** **space**. Beaula and Gifta [7] defined **2**-fuzzy **normed** **linear** **space** and proved some standard results. They also defined **2**-fuzzy dual **space** and proved it is complete **2**-fuzzy **normed** **linear** **space**. In section **2**, we recall some preliminary concepts and in section 3, we define **2**-fuzzy continuity and boundedness and a theorem is established related to these concepts. In section 4, the Uniform Boundedness theorem is established and in section 5, we introduce second fuzzy dual for the **2**-fuzzy **normed** **linear** **space**. Banach Alaoglu theorem is developed in this **space**.

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In [10] Iqbal H. Jebril and Samanta introduced fuzzy anti-norm on a **linear** **space** depending on the idea of fuzzy anti- norm was introduced by Bag and Samanta [3] and investigated their important properties. In [17,18] Surender Reddy introduced the notion of convergent sequence and Cauchy sequence in fuzzy anti-**2**-**normed** **linear** **space** and fuzzy anti- n-**normed** **linear** spaces. Recently Surender Reddy [19,20] studied on the set of all t-best approximations on fuzzy anti- **normed** **linear** spaces and fuzzy anti-**2**-**normed** **linear** spaces.

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• , • is called a **2**-norm on X and the pair (X, • , • ) is called a **linear** **2**-**normed** **space**. Definition 2.2 [11]. Let n ∈ N (natural numbers) and let X be a real vector **space** of di- mension d ≥ n. (Here we allow d to be infinite.) A real-valued function • ,. . ., • on X × ··· × X

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with a unique fixed point. In [10], he narrowed the quasi-**normed** **linear** **space** to quasi-**2**- **normed** **linear** **space** Q-**2**-NLS. Addition properties of the Q-**2**-NLS such as the convergence of the two Cauchy sequences and a pseudo-quasi-**2**-norm of the Q-**2**-NLS were observed by the authors in [11].

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Before proceeding with the main results, we recall some of the basic notations and definitions that are used in this paper. The notion of **2**–**normed** **space** was initially introduced by S. GÄahler [1] as an interesting **linear** generalization of a **normed** **linear** **space**, which was further studied by J.A. White and Y.J. Cho [**2**], K.Iseki [3], R. Freese et al. [4,5], W.Raymond et al. [6] and many others. Recently a lot of activities have been started by many researchers to study this concept in different directions, for instances E. Savas [7], H. Gunawan and H. Mashadi [8] , J.K. Srivastava and N.P. Pahari [9] , M. Açikgöz [10] and others.

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Recently, Cho et al. [] investigated the Mazur-Ulam theorem on probabilistic - **normed** spaces. Moslehian and Sadeghi [] investigated the Mazur-Ulam theorem in non-Archimedean spaces. Choy and Ku [] proved that the barycenter of a triangle car- ries the barycenter of a corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean -**normed** spaces using the above statement. Chen and Song [] introduced the concept of weak n-isometry, and then they got that under some condi- tions a weak n-isometry is also an n-isometry. Alaca [] gave the concepts of -isometry, collinearity, -Lipschitz mapping in -fuzzy -**normed** **linear** spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a -fuzzy -**normed** **linear** **space** or (X) is a fuzzy -**normed** **linear** **space**. Park and Alaca [] introduced the concept of -fuzzy n-**normed** **linear** **space** or fuzzy n-**normed** **linear** **space** of the set of all fuzzy sets of a non-empty set. They deﬁned the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this **space**. Also, they generalized the Mazur-Ulam theorem, that is, when X is a -fuzzy n-**normed** **linear** **space** or (X) is a fuzzy n-**normed** **linear** **space**, the Mazur- Ulam theorem holds. Moreover, it is shown that each n-isometry in -fuzzy n-**normed** **linear** spaces is aﬃne. Ren [] showed that every generalized area n preserving mapping between real -**normed** **linear** spaces X and Y which is strictly convex is aﬃne under some conditions.

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Sinha, Lal and Mishra [12] introduced the concept of fuzzy **2**-bounded **linear** operator on a fuzzy **2**-**normed** **linear** **space** to another fuzzy **2**-**normed** **linear** **space** and two types of (strong and weak) fuzzy **2**-bounded **linear** operators were also defined. They also discussed the relation between strong fuzzy **2**-bounded **linear** operator and weak fuzzy **2**-bounded **linear** operator.

The theory of fuzzy sets was introduced by L. Zadeh [9] in 1965. A satisfactory theory of **2**-norm on a **linear** **space** has been introduced and developed by Gahler in 1964[**2**]. The concepts fuzzy norm and α-norm were introduced by Bag and Samanta in 2003[1]. Jialuzhang [3] has defined fuzzy **linear** **space** in a different way. The notion of **2**-fuzzy **2**- **normed** **linear** **space** of the set of all fuzzy sets of a set was introduced by R. M. Somasundaram and Thangaraj Beaula [6]. The concept of intuitionistic 2fuzzy **2**-**normed** **linear** **space** of the set of all fuzzy sets of a set was introduced by Thangaraj Beaula and D. Lilly Esthar Rani [7].

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Order unit **normed** **linear** spaces are a special type of regularly ordered **normed** li- near spaces and therefore the first section is a short collection of the fundamental results on this type of **normed** **linear** spaces. The connection between order unit **normed** **linear** spaces and base **normed** **linear** spaces within the category of regularly ordered **normed** **linear** spaces is described in Section **2**, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit **normed** **linear** **space**. It is shown that the original results on Banach limits are valid for a greater range.

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i) f(x+y) = f(x) +f(y) ii) f(αx) = αf(x) iii) -f(-x) ≤ f(x) (or) f(w) ≥ 0 for every w∈ 𝑊 𝑋 . The set of all **almost** **linear** functional defined on an **almost** **linear** **space** X is denoted by X # . 2.4: Norm on an **almost** **linear** **space**: A norm ||| ∙ ||| on an **almost** **linear** **space** X is a function satisfying the following conditions N 1 – N 3 .

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Gahler [4] introduced the theory of n-norm on a **linear** **space**. For a systematic development of n-**normed** **linear** **space** one may refer to [5, 6, 8, 9]. In [5], Hendra Gunawan and Mashadi have also discussed the Cauchy sequence and convergent sequence in n-**normed** **linear** **space**. A detailed theory of fuzzy **normed** **linear** **space** can be found in [1, **2**, 3, 7, 11]. In [10, 13], Vijayabalaji extended n-**normed** **linear** **space** to fuzzy n-**normed** **linear** **space** and complete fuzzy n- **normed** **linear** **space**.

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[**2**] Pathak H.K., Cho Y.J., Kang S.M. and Madharia B., compatible mapping of type (C) and Common fixed point theorem of Gregus type, demonstr. Math., 31 (3) 1998, 499-517. [3] Pathak H. K. and Khan M.S., compatible mapping of type (B) and common fixed point theorem of Gregus type, Czechoslovak Math. J., 45 (120) (1995), 685-698.

In real life, most of the problems are uncertain or imprecise. This is due to the lack of information required. In order to deal such situations Zadeh [7] in 1965 introduced fuzzy sets. In fuzzy set theory thereis a lack of parameterization tools which lead Molodostov [3] in 1999 to introduce soft set theory. Maji et al [**2**] in 2001 studied the combination of fuzzy set theory and soft set theory and gave a new concept called fuzzy soft set. This new notion widened the approach of soft set from crisp cases to fuzzy cases. Topological study on fuzzy soft set theory was initiated by Tanay and Kandemir in [6]. In 2013 Zadeh [1] coined fuzzy soft norm over a set and established the relationship between fuzzy soft norm and fuzzy norm over a set.

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The concept of fuzzy norm was first introduced by Katsaras [1] in the year 1984. After that, in 1992, Felbin [**2**] defined a fuzzy norm on a **linear** **space** with an associated metric of the Kaleva and Seikkala type [3]. Further developement in the notion of fuzzy norm took place in 1994, when Cheng and Moderson [4] gave the definition of fuzzy norm in another approach having an associated metric of the Kramosil and Michalek type [5] . Thereafter, following the definition of fuzzy norm by Cheng and Moderson [4], Bag and Samanta [6] introduced the concept of fuzzy norm in a different way.

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Some results of fuzzy complete fuzzy **normed** spaces were studied by Saadati and Vaezpour in 2005 (1). Properties of fuzzy bounded **linear** operators on a fuzzy **normed** **space** were investigated by Bag and Samanta in 2005 (**2**).The fuzzy **normed** **linear** **space** and its fuzzy topological structure were studied by Sadeqi and Kia in 2009 (3). Properties of fuzzy continuous operators on a fuzzy **normed** **linear** spaces were studied by Nadaban in 2015 (4). The definition of the fuzzy norm of a fuzzy bounded **linear** operator was introduced by Kider and Kadhum in 2017 (5). Fuzzy functional analysis is developed by the concepts of fuzzy norm and a large number of researches by different authors have been published for reference please see (6, 7, 8, 9, 10).

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/ defined on a subspace M ofa normed linear space can be extended to a bounded linear. functional F defined on the entire space and with norm equal to the norm of/ on M,that[r]

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