Orientador: Jorge Tulio Mujica Ascui / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T12:22:20Z (GMT). No. of bitstreams: 1 Araujo_MariaAngelica_M.pdf: 648625 bytes, checksum: 6d3a2ed1bef26a3213e505351e408696 (MD5) Previous issue date: 2014 / Resumo: O objetivo desta dissertação é apresentar e demonstrar alguns teoremas da Análise matemática, são eles, O Teorema de Aproximação de Weierstrass, o Teorema de Kakutani-Stone, os Teoremas de Stone-Weierstrass e o Teorema de Nachbin. Para demonstrá-los relembraremos algumas definições e resultados básicos da teoria de Análise e Topologia e abordaremos as demais ferramentas necessárias para suas respectivas demonstrações / Abstract: The aim of this dissertation is to present and prove some theorems of mathematical analysis, that are, the Weierstrass Approximation Theorem, the Kakutani-Stone Theorem, the Stone-Weierstrass Theorems and the Nachbin Theorem. To prove them we recall some basic definitions and results of analysis and topology and we discuss other tools that are necessary for their respective proofs / Mestrado / Matematica / Mestra em Matemática

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions 𝑓ₙ. Since 𝑤 is a contraction map in the compact metric space 𝐹(𝑋), 𝑤 has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function 𝑓. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in 𝐶[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in 𝐶[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under 𝑤 and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let 𝒩 and 𝕽 denote the natural numbers and the real numbers, respectively.

The concept of (b, s)-continuous functions in topological spaces is introduced and studied. Some of their characteristic properties are considered. Also we investigate the relationships between these classes of functions and other classes of functions.

The ring of real-valued continuous functions on a completely regular frame L is denoted by RL. As usual, βL denotes the Stone-Cech compactification of ˇ L. In the thesis we study ideals of RL induced by sublocales of βL. We revisit the notion of purity in this ring and use it to characterize basically disconnected frames. The socle of the ring RL is characterized as an ideal induced by the sublocale of βL which is the join of all nowhere dense sublocales of βL. A localic map f : L → M induces a ring homomorphism Rh: RM → RL by composition, where h: M → L is the left adjoint of f. We explore how the sublocale-induced ideals travel along the ring homomorphism Rh, to and fro, via expansion and contraction, respectively. The socle of a ring is the sum of its minimal ideals. In the literature, the socle of RL has been characterized in terms of atoms. Since atoms do not always exist in frames, it is better to express the socle in terms of entities that exist in every frame. In the thesis we characterize the socle as one of the types of ideals induced by sublocales. A classical operator invented by Gillman, Henriksen and Jerison in 1954 is used to create a homomorphism of quantales. The frames in which every cozero element is complemented (they are called P-frames) are characterized in terms of some properties of this quantale homomorphism. Also characterized within the category of quantales are localic analogues of the continuous maps of R.G. Woods that characterize normality in the category of Tychonoff spaces. / Mathematical Sciences / Ph. D. (Mathematics)

No description available.

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L. / Mathematical Sciences / Mathematics / D.Phil. (Mathematics)

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L. / Mathematical Sciences / D.Phil. (Mathematics)

No description available.

No presente trabalho, estudamos a propriedade da c0-extensão no contexto de espaços de funções contínuas denidas numa reta compacta e tomando valores em R. Nosso principal resultado é que se K é uma reta compacta, então todo subespaço fechado e com dual separável de C(K) possui a propriedade da c0-extensão em C(K) e portanto, o espaço C(K) tem a propriedade de Sobczyk. Também apresentamos uma caracterização das funções phi: K --> L contínuas, crescentes e sobrejetoras entre retas compactas para as quais a subálgebra de Banach phi*C(L) possui a propriedade da c0-extensão em C(K). / In this work, we study the c0-extension property in the context of spaces of continuous real-valued functions defined in a compact line. Our main result states that if K is a compact line, then every closed subspace of C(K) with separable dual has the c0-extension property in C(K) and therefore, the space C(K) has the Sobczyk property. We also present a characterization of the continuous order-preserving surjective maps phi : K --> L between compact lines such that the Banach subalgebra phi*C(L) has the c0-extension property in C(K).

Para um espaço localmente compacto de Hausdorff K e um espaço de Banach X, denotamos por C0(K,X) o espaço de todas as funções a valores em X contínuas sobre K que se anulam no infinito, munido da norma do supremo. No espírito do clássico teorema de Banach-Stone 1937, estabelecemos que se C0(K1,X) é isomorfo a C0(K2,X), onde X é um espaço de Banach de cotipo finito e tal que X é separável ou X* tem a propriedade de Radon-Nikodým, então ou K1 e K2 são ambos finitos ou K1 e K2 tem a mesma cardinalidade. Trata-se de uma extensão vetorial de um resultado de Cengiz 1978, o caso escalar X = R ou X = C. Demonstramos também que se K1 e K2 são intervalos compactos de ordinais e X é um espaço de Banach de cotipo finito, então a existência de um isomorfismo T de C(K1,X) em C(K2,X) com ||T||||T-1|| < 3 implica que uma certa soma topológica finita de K1 é homeomorfa a alguma soma topológica finita de K2. Mais ainda, se Xn não contém subespaço isomorfo a Xn+1 para todo n &isin; N, então K1 é homeomorfo a K2. Em outras palavras, obtemos um teorema tipo Banach-Stone vetorial que é uma extensão de um teorema de Gordon de 1970 e ao mesmo tempo uma extensão de um teorema de Behrends e Cambern de 1988. Mostramos que se existe um isomorfismo T de C(K1) em um subespaço de C(K2,X) com ||T||||T-1|| < 3, então a cardinalidade do &alpha;-ésimo derivado de K2 ou é finita ou é maior do que a cardinalidade do &alpha;-ésimo derivado de K1, para todo ordinal &alpha;. Em seguida, seja n um inteiro positivo, &Gamma; um conjunto infinito munido da topologia discreta e X um espaço de Banach de cotipo finito. Estabelecemos que se o n-ésimo derivado de K for não vazio, então a distância de Banach-Mazur entre C0(K,X) e C0(&Gamma;,X) é maior ou igual a 2n + 1. Também demonstramos que para quaisquer inteiros positivos n e k, a distância de Banach-Mazur entre C([1,&omega;nk],X) e C0(N,X) é exatamente 2n+1. Estes resultados fornecem extensões vetoriais para alguns teoremas de Cambern de 1970. Para um ordinal enumerável &alpha;, denotando por C(&alpha;) o espaço de Banach das funções contínuas no intervalo de ordinal [1, &alpha;], obtemos cotas superiores H(n, k) e cotas inferiores G(n, k) para as distâncias de Banach-Mazur entre os espaços C(&omega;) e C(&omega;nk), 1 < n, k < &omega;, verificando H(n, k) - G(n, k) < 2. Estas estimativas fornecem uma resposta para uma questão de Bessaga e Peczynski de 1960 sobre as distâncias de Banach-Mazur entre C(&omega;) e cada um dos espaços C(&alpha;), &omega;<&alpha;<&omega;&omega;. / For a locally compact Hausdorff space K and a Banach space X, we denote by C0(K,X) the space of X-valued continuous functions on K which vanish at infinity, endowed with the supremum norm. In the spirit of the classical 1937 Banach-Stone theorem, we prove that if C0(K1,X) is isomorphic to C0(K2,X), where X is a Banach space having finite cotype and such that X is separable or X* has the Radon-Nikodým property, then either K1 and K2 are finite or K1 and K2 have the same cardinality. It is a vector-valued extension of a 1978 Cengiz result, the scalar case X = R or X = C. We also prove that if K1 and K2 are compact ordinal spaces and X is Banach space having finite cotype, then the existence of an isomorphism T from C(K1,X) onto C(K2,X) with ||T||||T-1|| < 3 implies that some finite topological sum of K1 is homeomorphic to some finite topological sum of K2. Moreover, if Xn contains no subspace isomorphic to Xn+1 for every n &isin; N, then K1 is homeomorphic to K2. In other words, we obtain a vector-valued Banach-Stone theorem which is an extension of a 1970 Gordon theorem and at same time an improvement of a 1988 Behrends and Cambern theorem. We show that if there is an embedding T of a C(K1) into C(K2,X) with ||T||||T-1|| < 3, then the cardinality of the &alpha;-th derivative of K2 is either finite or greater than the cardinality of the &alpha;-th derivative of K1, for every ordinal &alpha;. Next, let n be a positive integer, &Gamma; an infinite set with the discrete topology and X is a Banach space having finite cotype. We prove that if the n-th derivative of K is not empty, then the Banach Mazur distance between C0(K,X) and C0(&Gamma;,X) is greater than or equal to 2n + 1. Thus, we also show that for every positive integers n and k, the Banach Mazur distance between C([1,&omega;nk],X) and C0(N,X) is exactly 2n+1. These results provide vector-valued versions of some 1970 Cambern theorems. For a countable ordinal &alpha;, writing C(&alpha;) for the Banach space of continuous functions on the interval of ordinal [1, &alpha;], we give lower bounds H(n, k) and upper bounds G(n, k) on the Banach- Mazur distances between C(&omega;) and C(&omega;nk), 1 < n, k < &omega;, such that H(n, k) - G(n, k) < 2. These estimates provide an answer to a 1960 Bessaga and Peczynski question on the Banach-Mazur distances between C(&omega;) and each of the C(&alpha;) spaces, &omega;<&alpha;<&omega;&omega;.