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 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 On strong proximinality in normed linear spaces

2016 ◽  
Vol 70 (1) ◽  
pp. 19
Author(s):  
Sahil Gupta ◽  
T. D. Narang
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Linear Spaces ◽  
Quotient Spaces ◽  
Normed Linear Spaces ◽  
Rotund Banach Space ◽  
Approximative Compactness

The paper deals with strong proximinality in normed linear spaces. It is proved that in  a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and  approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

Weak Cauchy sequences in normed linear spaces

1964 ◽  
Vol 60 (4) ◽  
pp. 817-819 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Linear Space ◽  
Cauchy Sequence ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Finite Dimensional ◽  
Cauchy Sequences ◽  
Convex Cover

It follows from the Krein-Milman theorem that (c0) is not isomorphic to the dual of a Banach space. Using a technique due to Banach ((4), page 194) we shall extend this result to show that if a subspace of (c0) is isomorphic to the dual of a normed linear space, then it is finite dimensional (Proposition 1). Using this result, we shall show that if E is a normed linear space, the unit ball of which is contained in the closed absolutely convex cover of a weak Cauchy sequence, then Eis finite dimensional (Proposition 2). This result has applications to the Banach-Dieudonné theorem, and to the theory of two-norm spaces.

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 Product-Normed Linear Spaces

2018 ◽  
Vol 11 (3) ◽  
pp. 740-750
Author(s):  
Benedict Barnes ◽  
I. A. Adjei ◽  
S. K. Amponsah ◽  
E. Harris
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Linear Space ◽  
Linear Algebra ◽  
Functional Properties ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Space Product

In this paper, both the product-normed linear space $P-NLS$ (product-Banach space) and product-semi-normed linear space (product-semi-Banch space) are introduced. These normed linear spaces are endowed with the first and second product inequalities, which have a lot of applications in linear algebra and differential equations. In addition, we showed that $P-NLS$ admits functional properties such as completeness, continuity and the fixed point.

An orthogonality in normed linear spaces based on angular distance inequality

2015 ◽  
Vol 90 (2) ◽  
pp. 281-297 ◽  
Author(s):  
F. Dadipour ◽  
F. Sadeghi ◽  
A. Salemi
Keyword(s):  
Angular Distance ◽  
Linear Spaces ◽  
Normed Linear Spaces
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