Linear Spaces and Linear Operators

1986 ◽  
pp. 15-38
Author(s):  
Allan M. Krall
Keyword(s):  
Linear Operators ◽  
Linear Spaces

scholarly journals Continuity of Bounded Linear Operators on Normed Linear Spaces

2018 ◽  
Vol 26 (3) ◽  
pp. 231-237
Author(s):  
Kazuhisa Nakasho ◽  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Natural Extension ◽  
Linear Operators ◽  
Bilinear Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Lipschitz Continuous ◽  
Normed Linear Spaces ◽  
Whole Space ◽  
System 1

Summary In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.

scholarly journals On numerical ranges of operators on locally convex spaces

1975 ◽  
Vol 20 (4) ◽  
pp. 468-482 ◽  
Author(s):  
J. R. Giles ◽  
G. Joseph ◽  
D. O. Koehler ◽  
B. Sims
Keyword(s):  
Numerical Range ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Range Theory ◽  
Locally Convex ◽  
Convex Spaces

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moore's work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.

scholarly journals Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces

2015 ◽  
Vol 10 (6) ◽  
pp. 74 ◽  
Author(s):  
Sorin Nadaban
Keyword(s):  
Continuous Mapping ◽  
Uniform Boundedness ◽  
Uniform Continuity ◽  
Linear Operators ◽  
Closed Graph Theorem ◽  
Linear Spaces ◽  
Open Mapping Theorem ◽  
Normed Linear Spaces ◽  
Continuous Mappings ◽  
Uniform Boundedness Principle

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

scholarly journals Some generalizations of ascent and descent for linear operators

2021 ◽  
Author(s):  
Zied Garbouj
Keyword(s):  
Linear Operator ◽  
Linear Space ◽  
Positive Operator ◽  
Finite Rank ◽  
Operator Theory ◽  
Compact Operators ◽  
Linear Operators ◽  
Linear Spaces ◽  
Basic Properties ◽  
Stability Results

AbstractThe purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

Fuzzy Bounded Linear Operator in Fuzzy Normed Linear Spaces and its Fuzzy Compactness

2020 ◽  
Vol 16 (01) ◽  
pp. 177-193
Author(s):  
Mami Sharma ◽  
Debajit Hazarika
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
The Relationship

In this paper, we first investigate the relationship between various notions of fuzzy boundedness of linear operators in fuzzy normed linear spaces. We also discuss the fuzzy boundedness of fuzzy compact operators. Furthermore, the spaces of fuzzy compact operators have been studied.

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