In this study, we examine some important subspaces by showing that the set of n-dimensional inter... more In this study, we examine some important subspaces by showing that the set of n-dimensional interval vectors is a quasilinear space. By defining the concept of dimensions in these spaces, we show that the set of $n$-dimensional interval vectors is actually a $(n_{r},n_{s})$-dimensional quasilinear space and any quasilinear space is $\left( n_{r},0_{s}\right) $-dimensional if and only if it is $n$-dimensional linear space. We also give examples of $(2_{r},0_{s})$ and $(0_{r},2_{s})$-dimensional subspaces. We define the concept of dimension in a quasilinear space with natural number pairs. Further, we define an inner product on some spaces and talk about them as inner product quasilinear spaces. Further, we show that some of them have Hilbert quasilinear space structure.
Fundamentals of Contemporary Mathematical Sciences
This paper presents some important classes of the continuous functions defined from the set of re... more This paper presents some important classes of the continuous functions defined from the set of real numbers to the space of complex intervals. These function spaces have an algebraic structure named as a quasilinear space which is suggested by Aseev in 1986. In this work, we analysis the quasilinear structure on the classes of the continuous and complex interval-valued functions. Further, we show that these spaces are the normed Ω-spaces. Finally, we examine the dimension of these function spaces.
It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, va... more It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F ℝ n and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, u q = ∫ 0 1 sup x ∈ u α x q d α 1 / q where 1 ≤ q ≤ ∞ , we stated that F ℝ n , u q is a complete normed space. Furthermore, we introduced an inner product in this space for the case q = 2 . The inner product must be in the form u , v = ∫ 0 1 u α , v α K ℝ n d α = ∫ 0 1 a , b ℝ n d α : a ∈ u α , b ∈ v α . For u , v ∈ F ℝ n . We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F ℝ n . Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasiline...
Nonlinear functional analysis and applications, 2016
In this paper concerning with the notion of convergence in normed quasilinear spaces, we show tha... more In this paper concerning with the notion of convergence in normed quasilinear spaces, we show that convergence of a sequence depends directly on partial order relation defined on the normed quasilinear space. Then we introduce the notions of “lower semi convergence” and “upper semi convergence”. We note that these new definitions about convergence would be an alternative for not convergent sequences in normed quasilinear spaces.
Aseev lineer uzaylarin daha genel bir formu olan quasilineer uzay kavramini tanimladi, [1]. Quasi... more Aseev lineer uzaylarin daha genel bir formu olan quasilineer uzay kavramini tanimladi, [1]. Quasilineer uzaylar teorisindeki temel bir eksiklik, lineer bagimlilik-bagimsizlik ve baz kavramlarinin yoklugudur. Belki de bu, quasilineer uzaylar teorisinin gelisimi onundeki en buyuk engeldir. Bu calismada, bu onemli kavramlarin elde ettigimiz yeni tanimlarini sunacagiz. Verdigimiz tanimlarin, lineer uzaylara iliskin benzeri sonuclarla tutarlilik icinde verildigini de gosterecegiz. Arastirmalarimiz bu kavramlarin direkt olarak quasilineer uzaylardaki siralama bagintisina bagli oldugunu ve bu konjonktur icin, lineer bagimlilik-bagimsizlik tanimini alt ve ust quasilineer bagimlilik-bagimsizlik gibi iki parcada sunmamiz gerektigini gostermektedir. Bu nedenle calismanin son bolumunde, oncelikle bir quasilineer uzayda sonlu bir {x_{k}}_{k=1}? kumesinin alt ve ust quasilineer kombinasyonu ve alt ve ust quasilineer bagimsizligi, daha sonra ayni kumenin alt ve ust gereni tanitilmistir. Bu temelle...
We introduced and investigated the sequence space \ (X¡¡,r, f,s) defined by a modulus function / ... more We introduced and investigated the sequence space \ (X¡¡,r, f,s) defined by a modulus function / , and constructed its FK-structure under some conditions. Also we exposed some inclusion relations among the variations of the space.
In this paper, we state some properties of inner pr oduct quasilinear spaces. Moreover we introdu... more In this paper, we state some properties of inner pr oduct quasilinear spaces. Moreover we introduce the concepts of inner product Ω-space, Hilbert Ω-space and investigate some related theorems. Also, we establi h some differences of inner product quasilinear spaces from the inner product (linear) spaces.
Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalu... more Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math., 2 (1986), 23-52] generalized linear spaces by introducing the notion of quasilinear spaces in 1986. Then, special quasilinear spaces which are called "solid floored quasilinear spaces" were defined and their some properties examined in [ C ¸akan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, ˙In ön ü University, Malatya, 2016]. In fact, this classification was made so as to examine consistent and detailed some properties related to quasilinear spaces. In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. At the same time, the mentioned classification is crucial for define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems. Also, we try to explain some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.
In this paper, we investigate a very important function space consists of set-valued functions de... more In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear ...
Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2020
Bu çalışmada quasilineer fonksiyonel analizde yeni bir kavram olan biquasilineer fonksiyonel kavr... more Bu çalışmada quasilineer fonksiyonel analizde yeni bir kavram olan biquasilineer fonksiyonel kavramını tanımladık. Bilineer fonksiyonel kavramında olduğu gibi biquasilineer fonksiyonel ve kuadratik form kavramlarında da bir iç çarpım quasilineer uzayına ihtiyaç duyulmadığını gördük. Bu fonksiyonelleri herhangi bir quasilineer uzayında tanımlayabildik. Çalışmamızda bu yeni kavramı verdikten sonra Hilbert quasilineer uzaylarda biquasilineer fonksiyoneller teorisi üzerine dikkate değer bazı örnekler verdik. Ve yine bu teori üzerine bazı teoremler ve ispatlarını çalışmamızda sunduk.
In this paper we focus on developing bounded quasilinear interval-valued functions. We deal with ... more In this paper we focus on developing bounded quasilinear interval-valued functions. We deal with the Hahn-Banach extension theorem for interval-valued functions. Finally, we show that there are enough bounded quasilinear functionals on the space of interval-valued functions to distinguish between its the regular elements.
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2017
started a new field in functional analysis by introducing the concept of normed quasilinear space... more started a new field in functional analysis by introducing the concept of normed quasilinear spaces which is a generalization of classical normed linear spaces. Then, we introduced the normed proper quasilinear spaces in addition to the notions of regular and singular dimension of a quasilinear space, C ¸akan and Yılmaz (J Nonlinear Sci Appl 8: 2015). In this study, we classify the normed proper quasilinear spaces as ''solid-floored'' and ''non solid-floored''. Thus, some properties of normed proper quasilinear spaces become more comprehensible. Also we present the counterpart of classical Riesz lemma in normed quasilinear spaces.
In this study, we examine some important subspaces by showing that the set of n-dimensional inter... more In this study, we examine some important subspaces by showing that the set of n-dimensional interval vectors is a quasilinear space. By defining the concept of dimensions in these spaces, we show that the set of $n$-dimensional interval vectors is actually a $(n_{r},n_{s})$-dimensional quasilinear space and any quasilinear space is $\left( n_{r},0_{s}\right) $-dimensional if and only if it is $n$-dimensional linear space. We also give examples of $(2_{r},0_{s})$ and $(0_{r},2_{s})$-dimensional subspaces. We define the concept of dimension in a quasilinear space with natural number pairs. Further, we define an inner product on some spaces and talk about them as inner product quasilinear spaces. Further, we show that some of them have Hilbert quasilinear space structure.
Fundamentals of Contemporary Mathematical Sciences
This paper presents some important classes of the continuous functions defined from the set of re... more This paper presents some important classes of the continuous functions defined from the set of real numbers to the space of complex intervals. These function spaces have an algebraic structure named as a quasilinear space which is suggested by Aseev in 1986. In this work, we analysis the quasilinear structure on the classes of the continuous and complex interval-valued functions. Further, we show that these spaces are the normed Ω-spaces. Finally, we examine the dimension of these function spaces.
It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, va... more It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F ℝ n and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, u q = ∫ 0 1 sup x ∈ u α x q d α 1 / q where 1 ≤ q ≤ ∞ , we stated that F ℝ n , u q is a complete normed space. Furthermore, we introduced an inner product in this space for the case q = 2 . The inner product must be in the form u , v = ∫ 0 1 u α , v α K ℝ n d α = ∫ 0 1 a , b ℝ n d α : a ∈ u α , b ∈ v α . For u , v ∈ F ℝ n . We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F ℝ n . Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasiline...
Nonlinear functional analysis and applications, 2016
In this paper concerning with the notion of convergence in normed quasilinear spaces, we show tha... more In this paper concerning with the notion of convergence in normed quasilinear spaces, we show that convergence of a sequence depends directly on partial order relation defined on the normed quasilinear space. Then we introduce the notions of “lower semi convergence” and “upper semi convergence”. We note that these new definitions about convergence would be an alternative for not convergent sequences in normed quasilinear spaces.
Aseev lineer uzaylarin daha genel bir formu olan quasilineer uzay kavramini tanimladi, [1]. Quasi... more Aseev lineer uzaylarin daha genel bir formu olan quasilineer uzay kavramini tanimladi, [1]. Quasilineer uzaylar teorisindeki temel bir eksiklik, lineer bagimlilik-bagimsizlik ve baz kavramlarinin yoklugudur. Belki de bu, quasilineer uzaylar teorisinin gelisimi onundeki en buyuk engeldir. Bu calismada, bu onemli kavramlarin elde ettigimiz yeni tanimlarini sunacagiz. Verdigimiz tanimlarin, lineer uzaylara iliskin benzeri sonuclarla tutarlilik icinde verildigini de gosterecegiz. Arastirmalarimiz bu kavramlarin direkt olarak quasilineer uzaylardaki siralama bagintisina bagli oldugunu ve bu konjonktur icin, lineer bagimlilik-bagimsizlik tanimini alt ve ust quasilineer bagimlilik-bagimsizlik gibi iki parcada sunmamiz gerektigini gostermektedir. Bu nedenle calismanin son bolumunde, oncelikle bir quasilineer uzayda sonlu bir {x_{k}}_{k=1}? kumesinin alt ve ust quasilineer kombinasyonu ve alt ve ust quasilineer bagimsizligi, daha sonra ayni kumenin alt ve ust gereni tanitilmistir. Bu temelle...
We introduced and investigated the sequence space \ (X¡¡,r, f,s) defined by a modulus function / ... more We introduced and investigated the sequence space \ (X¡¡,r, f,s) defined by a modulus function / , and constructed its FK-structure under some conditions. Also we exposed some inclusion relations among the variations of the space.
In this paper, we state some properties of inner pr oduct quasilinear spaces. Moreover we introdu... more In this paper, we state some properties of inner pr oduct quasilinear spaces. Moreover we introduce the concepts of inner product Ω-space, Hilbert Ω-space and investigate some related theorems. Also, we establi h some differences of inner product quasilinear spaces from the inner product (linear) spaces.
Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalu... more Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math., 2 (1986), 23-52] generalized linear spaces by introducing the notion of quasilinear spaces in 1986. Then, special quasilinear spaces which are called "solid floored quasilinear spaces" were defined and their some properties examined in [ C ¸akan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, ˙In ön ü University, Malatya, 2016]. In fact, this classification was made so as to examine consistent and detailed some properties related to quasilinear spaces. In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. At the same time, the mentioned classification is crucial for define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems. Also, we try to explain some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.
In this paper, we investigate a very important function space consists of set-valued functions de... more In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear ...
Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2020
Bu çalışmada quasilineer fonksiyonel analizde yeni bir kavram olan biquasilineer fonksiyonel kavr... more Bu çalışmada quasilineer fonksiyonel analizde yeni bir kavram olan biquasilineer fonksiyonel kavramını tanımladık. Bilineer fonksiyonel kavramında olduğu gibi biquasilineer fonksiyonel ve kuadratik form kavramlarında da bir iç çarpım quasilineer uzayına ihtiyaç duyulmadığını gördük. Bu fonksiyonelleri herhangi bir quasilineer uzayında tanımlayabildik. Çalışmamızda bu yeni kavramı verdikten sonra Hilbert quasilineer uzaylarda biquasilineer fonksiyoneller teorisi üzerine dikkate değer bazı örnekler verdik. Ve yine bu teori üzerine bazı teoremler ve ispatlarını çalışmamızda sunduk.
In this paper we focus on developing bounded quasilinear interval-valued functions. We deal with ... more In this paper we focus on developing bounded quasilinear interval-valued functions. We deal with the Hahn-Banach extension theorem for interval-valued functions. Finally, we show that there are enough bounded quasilinear functionals on the space of interval-valued functions to distinguish between its the regular elements.
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2017
started a new field in functional analysis by introducing the concept of normed quasilinear space... more started a new field in functional analysis by introducing the concept of normed quasilinear spaces which is a generalization of classical normed linear spaces. Then, we introduced the normed proper quasilinear spaces in addition to the notions of regular and singular dimension of a quasilinear space, C ¸akan and Yılmaz (J Nonlinear Sci Appl 8: 2015). In this study, we classify the normed proper quasilinear spaces as ''solid-floored'' and ''non solid-floored''. Thus, some properties of normed proper quasilinear spaces become more comprehensible. Also we present the counterpart of classical Riesz lemma in normed quasilinear spaces.
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