Every continuous convex function defined on a separable Banach space is Gateaux differentiable on a dense G^ subset of the space E [Mazur]. Suppose we are given a sequence (xn) that Is dense in E. Can we always find a Gateaux differentiable point x such that x = z^=^anxn.for some sequence (an) with infinitely many non-zero terms so that Ση∞=1||anxn|| < co ? According to this paper, such points are called of "simple type," and shown to be dense in E. Mazur's theorem follows directly from the result and Rybakov's theorem (A countably additive vector measure F: E -* X on a cr-field is absolutely continuous with respect to |x*F] for some x* e Xs) can be shown without deep measure theoretic Involvement.

We present important results from Hilbert space and functional analysis for understanding the subject ofReproducing kernel Hilbert spaces. We then showcase the underlying theory and properties of Reproducingkernel Hilbert Spaces. Finally, we show how the theory of reproducing kernel Hilbert spaces is applicable inboth interpolation and machine learning.

In this paper we consider approximation schemes and questions of well-posedness for a general class of functional differential equations of neutral-type (NFDE) where the difference operator does not have an atom at zero. Equations of this type occur in the modeling of certain aeroelastic control problems and include many singular integro-differential equations. We obtain general necessary and sufficient conditions for the well-posedness of functional differential equations of neutral-type on the Banach-spaces RⁿxL_p. As an example of the well-posedness of the non-atomic NFDE-system that arises in the study of aeroelasticity is established on RⁿxL_p, 1≤p<2. Employing the equivalence between generalized solutions of NFDEs and mild solutions of the “corresponding” abstract Cauchy-problems, we make use of general approximation results for well-posed Cauchy-problems to establish and analyze the convergence of the “averaging projection” scheme on the Banach spaces RⁿxL_p, 1<p<∞, for a class of problems with atomic difference operators. / Ph. D.

Nesse trabalho introduzimos e desenvolvemos a teoria de hiper-ideais de aplicações multilineares contínuas e polinômios homogêneos contínuos entre espaços de Banach. A ideia central é refinar os conceitos de multi-ideais e de ideais de polinômios com o objetivo de explorar de forma mais aprofundada a natureza não-linear das aplicações envolvidas. Para isso tomamos a teoria de ideais de operadores lineares, aplicações multilineares e polinômios homogêneos, desenvolvida a partir dos trabalhos de Pietsch, tanto no caso linear como no caso multilinear, como referencial. Provamos resultados gerais para hiper-ideais, damos muitos exemplos ilustrativos, e desenvolvemos métodos para gerar hiper-ideais, tanto no caso multilinear como no caso polinomial. / In this work we introduce and develop the theory of hyper-ideals of multilinear mappings and homogeneous polynomials between Banach spaces. The main idea is to refine the concepts of multi-ideal and of ideal of polynomials with the purpose of exploring deeply the nonlinear nature of the underlying mappings. To do this we take the ideal theory of linear operators, multilinear mappings and homogeneous polynomials, developed from the works of Pietsch, both in the linear and nonlinear cases, as a reference. We prove general results for hyper-ideals, provide a number of illustrative examples, and develop methods to generate hyper-ideals of multilinear mappings, as well as of hyper-ideals of homogeneous polynomials.

Nesse trabalho introduzimos e desenvolvemos a teoria de hiper-ideais de aplicações multilineares contínuas e polinômios homogêneos contínuos entre espaços de Banach. A ideia central é refinar os conceitos de multi-ideais e de ideais de polinômios com o objetivo de explorar de forma mais aprofundada a natureza não-linear das aplicações envolvidas. Para isso tomamos a teoria de ideais de operadores lineares, aplicações multilineares e polinômios homogêneos, desenvolvida a partir dos trabalhos de Pietsch, tanto no caso linear como no caso multilinear, como referencial. Provamos resultados gerais para hiper-ideais, damos muitos exemplos ilustrativos, e desenvolvemos métodos para gerar hiper-ideais, tanto no caso multilinear como no caso polinomial. / In this work we introduce and develop the theory of hyper-ideals of multilinear mappings and homogeneous polynomials between Banach spaces. The main idea is to refine the concepts of multi-ideal and of ideal of polynomials with the purpose of exploring deeply the nonlinear nature of the underlying mappings. To do this we take the ideal theory of linear operators, multilinear mappings and homogeneous polynomials, developed from the works of Pietsch, both in the linear and nonlinear cases, as a reference. We prove general results for hyper-ideals, provide a number of illustrative examples, and develop methods to generate hyper-ideals of multilinear mappings, as well as of hyper-ideals of homogeneous polynomials.

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

This paper deals with the symmetric approximation numbers as well as the other types of s-numbers. Concerning the s-numbers in the Banach spaces, namely the app- roximation numbers the Kolmogorov numbers and the Gelfand numbers, we present a few of possible definitions and some of their properties. We present the symmetric approximation numbers and their relation to the other s-numbers. We also focus on the s-numbers in the quasi-Banach spaces. The situation is a bit different, as we can not use the Hahn-Banach Theorem. Therefore some of the previous definitions and properties can not be retained. Moreover we define the symmetric approximation num- bers in the quasi-Banach spaces and discuss the problematics of this definition. Finally, we deal with the Carl's inequality regarding the entropy numbers and the s-numbers. We derive the proof for the symmetric approximation numbers in both Banach and quasi-Banach case. 1

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-16T13:04:43Z No. of bitstreams: 1 arquivototal.pdf: 1098275 bytes, checksum: 59e9bc1ca11ff8f14d5629548ccaaaf8 (MD5) / Made available in DSpace on 2017-08-16T13:04:43Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1098275 bytes, checksum: 59e9bc1ca11ff8f14d5629548ccaaaf8 (MD5) Previous issue date: 2016-03-04 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we study a general version of the Pietsch Domination Theorem, due to Pellegrino, Santos and Seoane-Sep´ulveda, that improves the unified version present in [3] and recovers known Pietsch Domination-type theorems where the unified approach seems not to work. / Neste trabalho estudamos uma vers˜ao geral do Teorema da Domina¸c˜ao de Pietsch, devido a Pellegrino, Santos e Seoane-Sep´ulveda, que melhora a vers˜ao unificada presente em [3] e recupera conhecidos teoremas de domina¸c˜ao do tipo Pietsch onde a abordagem unificada parece n˜ao funcionar.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-02T10:00:56Z No. of bitstreams: 1 fabriciovieiraoliveira.pdf: 1771957 bytes, checksum: 8c6b8d555c008276c082082c55021a5e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-02T11:42:53Z (GMT) No. of bitstreams: 1 fabriciovieiraoliveira.pdf: 1771957 bytes, checksum: 8c6b8d555c008276c082082c55021a5e (MD5) / Made available in DSpace on 2016-02-02T11:42:53Z (GMT). No. of bitstreams: 1 fabriciovieiraoliveira.pdf: 1771957 bytes, checksum: 8c6b8d555c008276c082082c55021a5e (MD5) Previous issue date: 2014-04-24 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho tem por objetivo apresentar aplicações da teoria de ultraprodutos em Análise Funcional em espaços de Banach, especificamente nos problemas de extensão de funções holomorfas, constantes de polarização e ideais de operadores maximais. Também é realizada uma revisão dos conceitos relacionados a topologia, aplicações multilineares, ultrafiltros e ultraprodutos de espaços de Banach. / This is work aims to present a application of the ultraproducts theory in Functional Analysis in Banach spaces, specifically in the problems of extension of holomorphic functions, polarization constants and maximal operator ideals. Also is performed a review of concepts about topology, multilinear maps, ultrafilters and ultraproducts in Banach spaces.