Closed Graph Theorem and Open Mapping Theorem

2020 ◽  
pp. 133-148
Author(s):  
P.N. Natarajan
Keyword(s):  
Open Mapping ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Open Mapping 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 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.    

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.

B(?)-spaces and the closed graph theorem

1964 ◽  
Vol 153 (4) ◽  
pp. 293-298 ◽  
Author(s):  
Taqdir Husain
Keyword(s):  
Closed Graph ◽  
Graph Theorem ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document