scholarly journals Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space

2020 ◽  
Vol 25 (4) ◽  
pp. 32-39
Author(s):  
Raghad I. Sabri
Keyword(s):  
Linear Operator ◽  
Fuzzy Set ◽  
Open Mapping ◽  
Normed Spaces ◽  
Closed Graph ◽  
Closed Graph Theorem ◽  
Open Mapping Theorem ◽  
Length Space ◽  
Definition Of ◽  
First Time

The theory of fuzzy set includes many aspects that regard important and significant in different fields of science and engineering in addition to there applications. Fuzzy metric and fuzzy normed spaces are essential structures in the fuzzy set theory. The concept of fuzzy length space has been given analogously and the properties of this space are studied few years ago. In this work, the definition of a fuzzy open linear operator is presented for the first time and the fuzzy Barise theorem is established to prove the fuzzy open mapping theorem in a fuzzy length space. Finally, the definition of a fuzzy closed linear operator on fuzzy length space is introduced to prove the fuzzy closed graph theorem.    

scholarly journals The closed graph theorem without category

1987 ◽  
Vol 36 (2) ◽  
pp. 283-287 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Baire Category ◽  
Baire Category Theorem ◽  
Normed Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Category Theorem

We show that a diagonal theorem of P. Antosik can be used to give a proof of the Closed Graph Theorem for normed spaces which does not depend upon the Baire Category Theorem.

scholarly journals An Open Mapping Theorem For Pro-Lie Groups

2007 ◽  
Vol 83 (1) ◽  
pp. 55-78 ◽  
Author(s):  
Karl H. Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Lie Groups ◽  
Projective Limit ◽  
Factor Group ◽  
Continuous Group ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Open Mapping Theorem ◽  
Finite Dimensional ◽  
Group Modulo

AbstractA pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

scholarly journals Banach’s Continuous Inverse Theorem and Closed Graph Theorem

2012 ◽  
Vol 20 (4) ◽  
pp. 271-274 ◽  
Author(s):  
Hideki Sakurai ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Operator Theory ◽  
Closed Linear Operator ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Text Book ◽  
Standard Text ◽  
Continuous Inverse

Summary In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.

scholarly journals Quotient operators and the open mapping theorem

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6221-6227
Author(s):  
Mahesh Krishna ◽  
Sam Johnson
Keyword(s):  
Open Mapping ◽  
Bounded Operators ◽  
Normed Spaces ◽  
Open Mapping Theorem

Quotients of bounded operators on normed spaces have been discussed. Openmapping theorem for quotients of bounded operators and its consequences are given.

scholarly journals Some applications of the open mapping theorem in locally convex cones

2021 ◽  
Vol 73 (3) ◽  
pp. 425-430
Author(s):  
S. Jafarizad ◽  
A. Ranjbari
Keyword(s):  
Linear Operator ◽  
Convex Cones ◽  
Open Mapping ◽  
Open Mapping Theorem ◽  
Locally Convex Cones ◽  
Locally Convex

UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).  

The Open Mapping and Closed Graph Theorem for Embeddable Topological Semigroups

1975 ◽  
Vol 18 (5) ◽  
pp. 671-674
Author(s):  
Douglass L. Grant
Keyword(s):  
Independent Interest ◽  
Open Mapping ◽  
Topological Groups ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Preliminary Results ◽  
Topological Semigroups

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.

Fuzziness: From Epistemic Considerations to Terminological Clarification

1997 ◽  
Vol 05 (04) ◽  
pp. 481-503 ◽  
Author(s):  
Herbert Toth
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Mathematical Concepts ◽  
Definition Of ◽  
First Time ◽  
Epistemic Role ◽  
Basic Philosophy

Using some semantical considerations and results from basic philosophy and some basic mathematical concepts we try to fix suitable explanations for some fundamental notions of fuzzy set theory on a semiformal level. This will – to the author's knowledge for the first time at all – lead us to a proposal for a definition of what is to be understood by the term 'fuzziness', simultaneously clarifying its epistemic rôle.

scholarly journals On the spectrum of a linear operator

1972 ◽  
Vol 13 (2) ◽  
pp. 98-101
Author(s):  
Michael B. Dollinger ◽  
Kirti K. Oberai
Keyword(s):  
Linear Operator ◽  
Compact Operator ◽  
Spectral Theory ◽  
Normed Spaces ◽  
Spectrum Of An Operator ◽  
Underlying Space ◽  
Definition Of ◽  
The Relationship

In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory.

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