scholarly journals Multilinear Operator and Its Basic Properties

2019 ◽  
Vol 27 (1) ◽  
pp. 35-45
Author(s):  
Kazuhisa Nakasho
Keyword(s):  
Algebraic Structure ◽  
Multilinear Operator ◽  
Multilinear Operators ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Basic Properties ◽  
Normed Linear Spaces ◽  
Real Linear

Summary In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.

scholarly journals Continuity of Multilinear Operator on Normed Linear Spaces

2019 ◽  
Vol 27 (1) ◽  
pp. 61-65
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Multilinear Operator ◽  
Multilinear Operators ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces

Summary In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism [4], [1] and [2]. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space. In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to [5], [11], [8], [9] in this formalization.

scholarly journals Dual Spaces and Hahn-Banach Theorem

2014 ◽  
Vol 22 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Extension Theorem ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Dual Spaces ◽  
Basic Properties ◽  
Banach Theorem ◽  
Real Linear

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].

scholarly journals Some Results on G-Normed Linear Spaces

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Maiada Nazar Mohammedali ◽  
Raghad Ibraham Sabri ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Normed Space ◽  
Cartesian Product ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Definition Of

In the present work, our goal is to define the Cartesian product of two generalized normed spaces depending on the notion of generalized normed space. It is a background to state and prove that the Cartesian product of two complete generalized normed spaces is also a complete generalized normed space. Furthermore, the definition of the pseudo-generalized normed space is introduced and essential concepts related to this space are discussed and proved.

scholarly journals Bilinear Operators on Normed Linear Spaces

2019 ◽  
Vol 27 (1) ◽  
pp. 15-23
Author(s):  
Kazuhisa Nakasho
Keyword(s):  
Banach Space ◽  
Algebraic Structure ◽  
Algebraic Structures ◽  
Linear Spaces ◽  
The Third ◽  
Bilinear Operators ◽  
Normed Linear Spaces

Summary The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar [1]. In the first two chapters, algebraic structures [3] of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space. In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to [4], [11], [2], [7] and [8] in this formalization.

scholarly journals On (fuzzy) pseudo-semi-normed linear spaces

2021 ◽  
Vol 7 (1) ◽  
pp. 467-477
Author(s):  
Yaoqiang Wu ◽  
Keyword(s):  
The Other ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Other Hand ◽  
Fuzzy Topological Spaces

<abstract><p>In this paper, we introduce the notion of pseudo-semi-normed linear spaces, following the concept of pseudo-norm which was presented by Schaefer and Wolff, and illustrate their relationship. On the other hand, we introduce the concept of fuzzy pseudo-semi-norm, which is weaker than the notion of fuzzy pseudo-norm initiated by N$ \tilde{\rm{a}} $d$ \tilde{\rm{a}} $ban. Moreover, we give some examples which are according to the commonly used $ t $-norms. Finally, we establish norm structures of fuzzy pseudo-semi-normed spaces and provide (fuzzy) topological spaces induced by (fuzzy) pseudo-semi-norms, and prove that the (fuzzy) topological spaces are (fuzzy) Hausdorff.</p></abstract>

scholarly journals ON I- CONVERGENCE OF SEQUENCES IN GRADUAL NORMED LINEAR SPACES

2021 ◽  
pp. 595
Author(s):  
Chiranjib Choudhury ◽  
Shyamal Debnath
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Basic Properties ◽  
Normed Linear Spaces ◽  
Cauchy Sequences ◽  
Convergence Of Sequences

In this paper, we introduce the concepts of $\mathcal{I}$ and $\mathcal{I^{*}}-$convergence of sequences in gradual normed linear spaces. We study some basic properties and implication relations of the newly defined convergence concepts. Also, we introduce the notions of $\mathcal{I}$ and $\mathcal{I^{*}}-$Cauchy sequences in the gradual normed linear space and investigate the relations between them.

scholarly journals SOME REMARKS ON ROUGH STATISTICAL \(\Lambda\)-CONVERGENCE OF ORDER \(\alpha\)

2021 ◽  
Vol 7 (1) ◽  
pp. 16
Author(s):  
Reena Antal ◽  
Meenakshi Chawla ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Linear Spaces ◽  
Basic Properties ◽  
Normed Linear Spaces ◽  
Bounded Sets ◽  
Convergent Sequences

The main purpose of this work is to define Rough Statistical \(\Lambda\)-Convergence of order \(\alpha\) \((0<\alpha\leq1)\) in normed linear spaces. We have proved some basic properties and also provided some examples to show that this method of convergence is more generalized than the rough statistical convergence. Further, we have shown the results related to statistically \(\Lambda\)-bounded sets of order \(\alpha\) and sets of rough statistically \(\Lambda\)-convergent sequences of order \(\alpha\).

A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Robert Skiba
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Differential Inclusions ◽  
Stationary Points ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Topological Invariance ◽  
Cohomological Index

AbstractWe construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.

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