In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

Neste trabalho estudamos algumas propriedades do espaço das funções contínuas munido da topologia da convergência pontual. Começamos estudando o espaço Cp(X) de forma geral, verificando que propriedades topológicas principais valem em Cp(X), usando teoremas de dualidade entre X e Cp(X). Em seguida estudamos a relação da estrutura topológica de X e a estrutura algébrica e topológica de Cp(X), onde o Teorema de Nagata é fundamental. Observamos algumas propriedades de X que são preservadas por l-equivalência ou t-equivalência, ou seja, que são determinadas pela estrutura linear topológica, ou pela estrutura topológica de Cp(X), respectivamente. Por último estudamos as condições para que Cp(X) seja um espaço de Lindelöf. Concluímos com a prova de Okunev de que o número de Lindelöf de Cp(X) é igual ao número de Lindelöf de Cp(X)xCp(X), para espaços fortemente zero-dimensionais X. / In this work we study some properties of the space of continuous functions endowed with the topology of pointwise convergence. We begin by studying the space Cp(X) in general terms, verifying that the main topological properties are valid in Cp(X), using duality theorems between X and Cp(X). Next we study the relationship between the topological structure of X and the algebraic as well as topological structure of Cp(X), in which the Nagata theorem theorem is essential. We observe some properties of X, which are preserved by l-equivalence or t-equivalence, i.e., which are respectively determined either by the linear topological structure of Cp(X) or by its topological one. Finally we study in which conditions Cp(X) is a Lindelöf space. We conclude with the proof of Okunev that the Lindelöf number of Cp(X) is equal to the Lindelöf number of Cp(X)xCp(X), for strongly zero-dimensional spaces X.

Neste trabalho estudamos algumas propriedades do espaço das funções contínuas munido da topologia da convergência pontual. Começamos estudando o espaço Cp(X) de forma geral, verificando que propriedades topológicas principais valem em Cp(X), usando teoremas de dualidade entre X e Cp(X). Em seguida estudamos a relação da estrutura topológica de X e a estrutura algébrica e topológica de Cp(X), onde o Teorema de Nagata é fundamental. Observamos algumas propriedades de X que são preservadas por l-equivalência ou t-equivalência, ou seja, que são determinadas pela estrutura linear topológica, ou pela estrutura topológica de Cp(X), respectivamente. Por último estudamos as condições para que Cp(X) seja um espaço de Lindelöf. Concluímos com a prova de Okunev de que o número de Lindelöf de Cp(X) é igual ao número de Lindelöf de Cp(X)xCp(X), para espaços fortemente zero-dimensionais X. / In this work we study some properties of the space of continuous functions endowed with the topology of pointwise convergence. We begin by studying the space Cp(X) in general terms, verifying that the main topological properties are valid in Cp(X), using duality theorems between X and Cp(X). Next we study the relationship between the topological structure of X and the algebraic as well as topological structure of Cp(X), in which the Nagata theorem theorem is essential. We observe some properties of X, which are preserved by l-equivalence or t-equivalence, i.e., which are respectively determined either by the linear topological structure of Cp(X) or by its topological one. Finally we study in which conditions Cp(X) is a Lindelöf space. We conclude with the proof of Okunev that the Lindelöf number of Cp(X) is equal to the Lindelöf number of Cp(X)xCp(X), for strongly zero-dimensional spaces X.