Diminnie, R. Freese  and many others developed new concept like linear2-normed space.The notion of an almostlinearspace (als) was introduced by G. Godini -. All spaces involved in this work are over the real field ℝ.S. Gahler, Y.J. Cho, C. S. Elumalai, R. Vijayaragavan - established some characterization of best approximation in terms of 2-semi inner products and normalized duality mapping associated with a linear2-normedspace. Basing on this we introduced a new concept called2-normedalmostlinear spaceand established some results of best approximation in 2- normedalmostlinearspace andsome results of best simultaneous approximation in 2- normedalmostlinearspace  . Mehmet A¸cıkg¨oz  introduced the concept𝜀-Approximation in generalized 2- normedlinearspace and established some results. Basing on this we introduced a new concept calledδ- Best approximation in 2-normedalmostlinearspace in this paper and we established some results on δ- best approximation and δ- orthogonalisation in 2- normedalmostlinearspace.
In 1965, the theory of fuzzy sets was introduced by L. Zadeh . In 1964, a satisfactory theory of 2-norm on a linearspace has been introduced and developed by Gahler . In 2003, the concepts fuzzy norm and α-norm were introduced by Bag and Samanta . Jialu Zhang  has defined fuzzy linearspace in a different way. The notion of 2-fuzzy 2-normedlinearspace of the set of all fuzzy sets of a set was introduced by R.M. Somasundaram and Thangaraj Beaula . The concept of intuitionistic 2fuzzy 2-normedlinearspace of the set of all fuzzy sets of a set was introduced by Thangaraj Beaula and Lilly Esthar Rani .
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  introduced the notion called a generalized metric in 1928. On the other hand Vulich  in 1938 defined a notion of higher dimensional norm in linear spaces. The concept of 2-normedspace was developed by Gahler in the middle of 1960’s . 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 . In 1965, Zadeh  introduced the concept of the fuzzy set in this seminal paper. Somasundaram and Beaula  have newly coined 2-fuzzy normedlinearspace 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-normedlinearspace and prove some interesting theorems. 2. Preliminaries
Recently, Somasundaram and Beaula  introduced the concept of 2-fuzzy 2- normedlinearspace or fuzzy 2-normedlinearspace of the set of all fuzzy sets of a set. The authors gave the notion of a-2-norm on a linearspace corresponding to the 2- fuzzy 2-norm by using some ideas of Bag and Samanta  and also gave some funda- mental properties of this space.
The concept of fuzzy set was introduced by Zadeh  in 1965. A satisfactory theory of 2-norm on a linearspace has been introduced and developed by Gahler in . 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 . Jialuzhang  has defined fuzzy linearspace in a different way. Bag and Samanta  defined fuzzy bounded linear operator and fuzzy bounded linear functional on the fuzzy dual space. Somasundaram and Beaula  defined the notion of 2-fuzzy 2-normedlinearspace. They also established the famous closed graph theorem and Riesz theorem in 2-fuzzy 2-normedlinearspace. Beaula and Gifta  defined 2-fuzzy normedlinearspace and proved some standard results. They also defined 2-fuzzy dual space and proved it is complete 2-fuzzy normedlinearspace. 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 normedlinearspace. Banach Alaoglu theorem is developed in this space.
In  Iqbal H. Jebril and Samanta introduced fuzzy anti-norm on a linearspace depending on the idea of fuzzy anti- norm was introduced by Bag and Samanta  and investigated their important properties. In [17,18] Surender Reddy introduced the notion of convergent sequence and Cauchy sequence in fuzzy anti-2-normedlinearspace and fuzzy anti- n-normedlinear spaces. Recently Surender Reddy [19,20] studied on the set of all t-best approximations on fuzzy anti- normedlinear spaces and fuzzy anti-2-normedlinear spaces.
• , • is called a 2-norm on X and the pair (X, • , • ) is called a linear2-normedspace. Definition 2.2 . 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
with a unique fixed point. In , he narrowed the quasi-normedlinearspace to quasi-2- normedlinearspace 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 .
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–normedspace was initially introduced by S. GÄahler  as an interesting linear generalization of a normedlinearspace, which was further studied by J.A. White and Y.J. Cho , K.Iseki , R. Freese et al. [4,5], W.Raymond et al.  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 , H. Gunawan and H. Mashadi  , J.K. Srivastava and N.P. Pahari  , M. Açikgöz  and others.
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 -normedlinear spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a -fuzzy -normedlinearspace or (X) is a fuzzy -normedlinearspace. Park and Alaca  introduced the concept of -fuzzy n-normedlinearspace or fuzzy n-normedlinearspace 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-normedlinearspace or (X) is a fuzzy n-normedlinearspace, the Mazur- Ulam theorem holds. Moreover, it is shown that each n-isometry in -fuzzy n-normedlinear spaces is aﬃne. Ren  showed that every generalized area n preserving mapping between real -normedlinear spaces X and Y which is strictly convex is aﬃne under some conditions.
Sinha, Lal and Mishra  introduced the concept of fuzzy 2-bounded linear operator on a fuzzy 2-normedlinearspace to another fuzzy 2-normedlinearspace 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  in 1965. A satisfactory theory of 2-norm on a linearspace has been introduced and developed by Gahler in 1964. The concepts fuzzy norm and α-norm were introduced by Bag and Samanta in 2003. Jialuzhang  has defined fuzzy linearspace in a different way. The notion of 2-fuzzy 2- normedlinearspace of the set of all fuzzy sets of a set was introduced by R. M. Somasundaram and Thangaraj Beaula . The concept of intuitionistic 2fuzzy 2-normedlinearspace of the set of all fuzzy sets of a set was introduced by Thangaraj Beaula and D. Lilly Esthar Rani .
Order unit normedlinear 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 normedlinear spaces. The connection between order unit normedlinear spaces and base normedlinear spaces within the category of regularly ordered normedlinear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normedlinearspace. It is shown that the original results on Banach limits are valid for a greater range.
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 almostlinear functional defined on an almostlinearspace X is denoted by X # . 2.4: Norm on an almostlinearspace: A norm ||| ∙ ||| on an almostlinearspace X is a function satisfying the following conditions N 1 – N 3 .
Gahler  introduced the theory of n-norm on a linearspace. For a systematic development of n-normedlinearspace one may refer to [5, 6, 8, 9]. In , Hendra Gunawan and Mashadi have also discussed the Cauchy sequence and convergent sequence in n-normedlinearspace. A detailed theory of fuzzy normedlinearspace can be found in [1, 2, 3, 7, 11]. In [10, 13], Vijayabalaji extended n-normedlinearspace to fuzzy n-normedlinearspace and complete fuzzy n- normedlinearspace.
 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.  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  in 1965 introduced fuzzy sets. In fuzzy set theory thereis a lack of parameterization tools which lead Molodostov  in 1999 to introduce soft set theory. Maji et al  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 . In 2013 Zadeh  coined fuzzy soft norm over a set and established the relationship between fuzzy soft norm and fuzzy norm over a set.
The concept of fuzzy norm was first introduced by Katsaras  in the year 1984. After that, in 1992, Felbin  defined a fuzzy norm on a linearspace with an associated metric of the Kaleva and Seikkala type . Further developement in the notion of fuzzy norm took place in 1994, when Cheng and Moderson  gave the definition of fuzzy norm in another approach having an associated metric of the Kramosil and Michalek type  . Thereafter, following the definition of fuzzy norm by Cheng and Moderson , Bag and Samanta  introduced the concept of fuzzy norm in a different way.
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 normedspace were investigated by Bag and Samanta in 2005 (2).The fuzzy normedlinearspace and its fuzzy topological structure were studied by Sadeqi and Kia in 2009 (3). Properties of fuzzy continuous operators on a fuzzy normedlinear 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).