This paper covers the notation and basic results of the theory of ordered abelian groups required to state Hahn's Theorem, which deals with the structure of such groups. In addition, we present a simple proof of Hahn's theorem for groups of nite range and some results concerning the minimality of the immersion sugested by the theorem. / Este trabajo presenta las notaciones y resultados básicos de la teoríıa de grupos abelianos ordenados necesarios para establecer el teorema de Hahn, el cual describe la estructura de este tipo de grupos. Además presentamos una prueba simple del teorema de Hahn para grupos ordenados de rango finito y algunos resultados sobre la minimalidad de la inmersión descrita en el teorema.

We define valuations and dominance relations in torsion-free abelian groups and prove that they are essentially the same objects. Next we show that valuations correspond with ltrations of subgroups closed under division by integers. We also prove that every torsion-free abelian valued group can be embedded in the Hahn product of subgroups defined by the respective valuations. / En este trabajo definimos valoraciones y relaciones de dominación en grupos abelianos sin torsión y probamos que estos son esencialmente los mismos objetos. Adicionalmente probamos que las valoraciones también se corresponden con filtraciones de subgrupos cerrados por división por enteros y que todo grupo abeliano valorado y sin torsión puede sumergirse en el producto de Hahn de subgrupos definidos por la valoración.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T14:00:23Z No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) / Made available in DSpace on 2017-08-17T14:00:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) Previous issue date: 2015-02-27 / In this work we investigate the concept of topological vector spaces and their properties. In the rst chapter we present two sections of basic results and in the other sections we present a more general study of such spaces. In the second chapter we restrict ourselves to the real scalar eld and we study, in the context of locally convex spaces, the Hahn-Banach and Banach-Alaoglu theorems. We also build the weak, weak-star, of bounded convergence and of pointwise convergence topologies. Finally we investigate the Theorem of Banach-Steinhauss, the Open Mapping Theorem and the Closed Graph Theorem. / Neste trabalho, estudamos o conceito de espa cos vetoriais topol ogicos e suas propriedades. No primeiro cap tulo, apresentamos duas se c~oes de resultados b asicos e, nas demais se c~oes, apresentamos um estudo sobre tais espa cos de forma mais ampla. No segundo cap tulo, restringimo-nos ao corpo dos reais e fazemos um estudo sobre os espa cos localmente convexos, o Teorema de Hahn-Banach, o Teorema de Banach- Alaoglu, constru mos as topologias fraca, fraca-estrela, da converg^encia limitada e da converg^encia pontual. Por ultimo, estudamos o Teorema da Limita c~ao Uniforme, o Teorema do Gr a co Fechado e o da Aplica c~ao Aberta no contexto mais geral dos espa cos de Fr echet.