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Master's Dissertation
DOI
10.11606/D.45.2017.tde-03042017-145643
Document
Author
Full name
Janaína Baldan Santos
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2016
Supervisor
Committee
Vieira, Daniela Mariz Silva (President)
Galego, Eloi Medina
Mendes, Cristiane de Andrade
Title in Portuguese
Variações do Teorema de Banach Stone
Keywords in Portuguese
Banach- Stone
Estrutura extremal
Funções contínuas
Abstract in Portuguese
Este trabalho tem por objetivo estudar algumas variações do teorema de Banach- Stone. Elas podem ser encontradas no artigo Variations on the Banach- Stone Theorem, [14]. Além disso, apresentamos um resultado, provado por D. Amir em [1], que generaliza a versão clássica do Teorema de Banach- Stone. Consideramos os espaços C(K) e C(L), que representam os espaços de funções contínuas de K em R e de L em R respectivamente, onde K e L são espaços Hausdor compactos. O enunciado da versão clássica do teorema de Banach- Stone é a seguinte: "Sejam K e L espaços Hausdor compactos. Então C(K) é isométrico a C(L) se e somente se, K e L são homeomorfos". Apresentamos a primeira das variações que considera isomorfismo entre álgebras e foi feita por Gelfand e Kolmogoro em [15], no ano de 1939. A segunda versão apresentada trata de isomorfismo isométrico e a demonstração é originalmente devida a Arens e Kelley e é encontrada em [2]. Finalmente, estudamos o teorema provado por D. Amir e apresentado em [1]. Este teorema generaliza o teorema clássico de Banach- Stone e tem o seguinte enunciado: Se K e L são espaços Hausdor compactos e T é um isomorfismo linear de C(K) sobre C(L), com ||T||.||T^||< 2 então K e L são homeomorfos
Title in English
Variations Banach- Stone Theorem
Keywords in English
Banach- Stone
Continuous functions
Extremal structure
Abstract in English
This work aims to study some variations of the Banach- Stone theorem. They can be found in the article Variations on the Banach- Stone Theorem, [14]. In addition, we present a result, proved by D. Amir in [1], that generalizes the classic version of the Theorem Banach- Stone. We consider the spacesC(K) andC(L), representing the spaces of continuous functions from K into R and from L into R respectively, where K and L are compact Hausdor spaces. The wording of the classic version of the Banach- Stone theorem is as follows: "Let K e L be compact Haudor spaces. Then C(K) isisometrictoC(L) if,andonlyif, K and L are homeomorphic".Here the first of the variations that considers isomorphism between algebras and was made by Gelfand and Kolmogoro in [15], in 1939. The second version presented is about isometric isomorphisms and the demonstration is originally due to Arens and Kelley and it is found in [2]. Finally, we study the theorem proved by D. Amir and presented in [1]. This theorem generalizes the classical theorem Banach- Stone and states the following: "Let K e L be compact Haudor spaces and let T be a linear isomorphism from C(K) into C(L), with ||T||.||T^||< 2. Then K and L are homeomorphic".
 
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Publishing Date
2017-04-11
 
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