2-normed almost linear space

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δ- Best Approximation in 2-Normed Almost Linear Space

δ- Best Approximation in 2-Normed Almost Linear Space

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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Fuzzy boundedness and contractiveness on intuitionistic 2-fuzzy 2-normed linear space

Fuzzy boundedness and contractiveness on intuitionistic 2-fuzzy 2-normed linear space

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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On Fuzzy Ward Continuity in an Intuitionistic 2-Fuzzy 2-Normed Linear Space

On Fuzzy Ward Continuity in an Intuitionistic 2-Fuzzy 2-Normed Linear Space

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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An introduction to 2 fuzzy n normed linear spaces and a new perspective to the Mazur Ulam problem

An introduction to 2 fuzzy n normed linear spaces and a new perspective to the Mazur Ulam problem

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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Uniform Boundedness Principle on 2-Fuzzy Normed Linear Spaces

Uniform Boundedness Principle on 2-Fuzzy Normed Linear Spaces

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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Vol 2, No 10 (2011)

Vol 2, No 10 (2011)

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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Fuzzy n normed linear space

Fuzzy n normed linear space

• , • 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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Product-Normed Linear Spaces

Product-Normed Linear Spaces

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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On 2 - normed space valued paranormed null sequence space

On 2 - normed space valued paranormed null sequence space

Before proceeding with the main results, we recall some of the basic notations and definitions that are used in this paper. The notion of 2normed 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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Mazur Ulam theorem under weaker conditions in the framework of 2 fuzzy 2 normed linear spaces

Mazur Ulam theorem under weaker conditions in the framework of 2 fuzzy 2 normed linear spaces

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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Vol 7, No 2 (2016)

Vol 7, No 2 (2016)

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.

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Vol 3, No 9 (2012)

Vol 3, No 9 (2012)

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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Banach Limits Revisited

Banach Limits Revisited

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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“Some Results on the Dual Space of Normed Almost Linear Space”

“Some Results on the Dual Space of Normed Almost Linear Space”

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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Vol 3, No 1 (2012)

Vol 3, No 1 (2012)

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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FIXED POINT RESULTS IN NORMED LINEAR SPACE

FIXED POINT RESULTS IN NORMED LINEAR SPACE

[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.

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A New Notion for Fuzzy Soft Normed Linear Space

A New Notion for Fuzzy Soft Normed Linear Space

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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Some fixed point results in fuzzy cone normed linear space

Some fixed point results in fuzzy cone normed linear space

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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Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces

Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces

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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A study of optimization in Hilbert Space

A study of optimization in Hilbert Space

/ 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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