Dissertations / Theses on the topic 'Geometry of Banach spaces'

The principal objects of study in this thesis are arbitrary spectral families, E, on a complex Banach space X. The central theme is the relationship between the geometry of X and the p-variation of E. We show that provided X is super- reflexive, then given any E, there exists a value 1 · p < 1, depending only on E and X, such that var p(E) < 1. If X is uniformly smooth we provide an explicit range of such values p, which depends only on E and the modulus of convexity of X*, delta X*(.).

Dans le premier chapitre, nous définissons la notion d’espaces à partitions pondérées qui généralise la structure d’espaces à murs mesurés et qui fournit un cadre géométrique à l’étude des actions isométriques affines sur des espaces de Banach pour les groupes localement compacts à base dénombrable. Dans un premier temps, nous caractérisons les actions isométriques affines propres sur des espaces de Banach en termes d’actions propres par automorphismes sur des espaces à partitions pondérées. Puis, nous nous intéressons aux structures de partitions pondérées naturelles pour les actions de certaines constructions de groupes : somme directe ; produit semi-directe ; produit en couronne et produit libre. Nous établissons ainsi des résultats de stabilité de la propriété PLp par ces constructions. Notamment, nous généralisons un résultat de Cornulier, Stalder et Valette de la façon suivante : le produit en couronne d’un groupe ayant la propriété PLp par un groupe ayant la propriété de Haagerup possède la propriété PLp. Dans le deuxième chapitre, nous nous intéressons aux espaces métriques quasi-médians - une généralisation des espaces hyperboliques à la Gromov et des espaces médians - et à leurs propriétés. Après l’étude de quelques exemples, nous démontrons qu’un espace δ-médian est δ′-médian pour tout δ′ ≥ δ. Ce résultat nous permet par la suite d’établir la stabilité par produit directe et par produit libre d’espaces métriques - notion que nous développons par la même occasion. Le troisième chapitre est consacré à la définition et l’étude d’une distance propre, invariante à gauche et qui engendre la topologie explicite sur les groupes localement compacts, compactement engendrés. Après avoir montré les propriétés précédentes, nous prouvons que cette distance est quasi-isométrique à la distance des mots sur le groupe et que la croissance du volume des boules est contrôlée exponentiellement
In the first chapter, we define the notion of spaces with labelled partitions which generalizes the structure of spaces with measured walls : it provides a geometric setting to study isometric affine actions on Banach spaces of second countable locally compact groups. First, we characterise isometric affine actions on Banach spaces in terms of proper actions by automorphisms on spaces with labelled partitions. Then, we focus on natural structures of labelled partitions for actions of some group constructions : direct sum ; semi-direct product ; wreath product and free product. We establish stability results for property PLp by these constructions. Especially, we generalize a result of Cornulier, Stalder and Valette in the following way : the wreath product of a group having property PLp by a Haagerup group has property PLp. In the second chapter, we focus on the notion of quasi-median metric spaces - a generalization of both Gromov hyperbolic spaces and median spaces - and its properties. After the study of some examples, we show that a δ-median space is δ′-median for all δ′ ≥ δ. This result gives us a way to establish the stability of the quasi-median property by direct product and by free product of metric spaces - notion that we develop at the same time. The third chapter is devoted to the definition and the study of an explicit proper, left-invariant metric which generates the topology on locally compact, compactly generated groups. Having showed these properties, we prove that this metric is quasi-isometric to the word metric and that the volume growth of the balls is exponentially controlled

In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the Assouad-Nagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed three-manifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed three-manifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a tree-graded quasi-isometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the Bestvina-Bromberg-Fujiwara quasi-trees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite Assouad-Nagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite Assouad-Nagata dimension if and only if each peripheral subgroup does.

Quelques aspects de la géométrie des espaces LipschitzEn premier lieu, nous donnons les propriétés fondamentales des espaces Lipschitz libres. Puis, nous démontrons que l'image canonique d'un espace métrique M est faiblement fermée dans l'espace libre associé F(M). Nous prouvons un résultat similaire pour l'ensemble des molécules.Dans le second chapitre, nous étudions les conditions sous lesquelles F(M) est isométriquement un dual. En particulier, nous généralisons un résultat de Kalton sur ce sujet. Par la suite, nous nous focalisons sur les espaces métriques uniformément discrets et sur les espaces métriques provenant des p-Banach.Au chapitre suivant, nous explorons le comportement de type l1 des espaces libres. Entre autres, nous démontrons que F(M) a la propriété de Schur dès que l'espace des fonctions petit-Lipschitz est 1-normant pour F(M). Sous des hypothèses supplémentaires, nous parvenons à plonger F(M) dans une somme l_1 d'espaces de dimension finie.Dans le quatrième chapitre, nous nous intéressons à la structure extrémale de $F(M)$. Notamment, nous montrons que tout point extrémal préservé de la boule unité d'un espace libre est un point de dentabilité. Si F(M) admet un prédual, nous obtenons une description précise de sa structure extrémale.Le cinquième chapitre s'intéresse aux fonctions Lipschitziennes à valeurs vectorielles. Nous généralisons certains résultats obtenus dans les trois premiers chapitres. Nous obtenons également un résultat sur la densité des fonctions Lipschitziennes qui atteignent leur norme
Some aspects of the geometry of Lipschitz free spaces.First and foremost, we give the fundamental properties of Lipschitz free spaces. Then, we prove that the canonical image of a metric space M is weakly closed in the associated free space F(M). We prove a similar result for the set of molecules.In the second chapter, we study the circumstances in which F(M) is isometric to a dual space. In particular, we generalize a result due to Kalton on this topic. Subsequently, we focus on uniformly discrete metric spaces and on metric spaces originating from p-Banach spaces.In the next chapter, we focus on l1-like properties. Among other things, we prove that F(M) has the Schur property provided the space of little Lipschitz functions is 1-norming for F(M). Under additional assumptions, we manage to embed F(M) into an l1-sum of finite dimensional spaces.In the fourth chapter, we study the extremal structure of F(M). In particular, we show that any preserved extreme point in the unit ball of a free space is a denting point. Moreover, if F(M) admits a predual, we obtain a precise description of its extremal structure.The fifth chapter deals with vector-valued Lipschitz functions.We generalize some results obtained in the first three chapters.We finish with some considerations of norm attainment. For instance, we obtain a density result for vector-valued Lipschitz maps which attain their norm

In this thesis Clarkson's inequalities and their generalizations are the main tools. The technique that can be used to prove Clarkson type inequalities in more dimensions is shown. We also establish Clarkson type inequalities in general Banach spaces and point out the connections between Clarkson's inequalities and the concept of type and cotype. The classical results on the von Neumann-Jordan constant, closely related to Clarkson's inequalities, are shortly presented. The concepts of moduli of convexity and smoothness, which are connected with the geometry of Banach spaces, are discussed. Some equivalent ways of describing modulus of convexity and some properties of this function are formulated. The estimation of the modulus of convexity for L(p)-spaces is presented as well. Finally, several examples of moduli of convexity and smoothness for different spaces are described.

Godkänd; 1999; 20070320 (ysko)

Dans les années 90, Gowers démontre un théorème de type Ramsey pour les bloc-suites dans les espaces de Banach, afin de prouver deux dichotomies d'espaces de Banach. Ce théorème, contrairement à la plupart des résultats de type Ramsey en dimension infinie, ne repose pas sur un principe des tiroirs, et en conséquence, sa formulation doit faire appel à des jeux. Dans une première partie de cette thèse, nous développons un formalisme abstrait pour la théorie de Ramsey en dimension infinie avec et sans principe des tiroirs, et nous démontrons dans celui-ci une version abstraite du théorème de Gowers, duquel on peut déduire à la fois le théorème de Mathias-Silver et celui de Gowers. On en donne à la fois une version exacte dans les espaces dénombrables, et une version approximative dans les espaces métriques séparables. On démontre également le principe de Ramsey adverse, un résultat généralisant à la fois le théorème de Gowers abstrait et la détermination borélienne des jeux dénombrables. On étudie aussi les limitations de ces résultats et leurs généralisations possibles sous des hypothèses supplémentaires de théorie des ensembles.Dans une seconde partie, nous appliquons les résultats précédents à la preuve de deux dichotomies d'espaces de Banach. Ces dichotomies ont une forme similaire à celles de Gowers, mais sont Hilbert-évitantes : elles assurent que le sous-espace obtenu n'est pas isomorphe à un espace de Hilbert. Ces dichotomies sont une nouvelle étape vers la résolution d'une question de Ferenczi et Rosendal, demandant si un espace de Banach séparable non-isomorphe à un espace de Hilbert possède nécessairement un grand nombre de sous-espaces, à isomorphisme près
In the 90's, Gowers proves a Ramsey-type theorem for block-sequences in Banach spaces, in order to show two Banach-space dichotomies. Unlike most infinite-dimensional Ramsey-type results, this theorem does not rely on a pigeonhole principle, and therefore it has to have a partially game-theoretical formulation. In a first part of this thesis, we develop an abstract formalism for Ramsey theory with and without pigeonhole principle, and we prove in it an abstract version of Gowers' theorem, from which both Mathias-Silver's theorem and Gowers' theorem can be deduced. We give both an exact version of this theorem in countable spaces, and an approximate version of it in separable metric spaces. We also prove the adversarial Ramsey principle, a result generalising both the abstract Gowers' theorem and Borel determinacy of countable games. We also study the limitations of these results and their possible generalisations under additional set-theoretical hypotheses. In a second part, we apply the latter results to the proof of two Banach-space dichotomies. These dichotomies are similar to Gowers' ones, but are Hilbert-avoiding, that is, they ensure that the subspace they give is not isomorphic to a Hilbert space. These dichotomies are a new step towards the solution of a question asked by Ferenczi and Rosendal, asking whether a separable Banach space non-isomorphic to a Hilbert space necessarily contains a large number of subspaces, up to isomorphism

No presente trabalho apresentamos dois teoremas obtidos por Gorak em 2011, que são generalizações para o Teorema de Banach-Stone, envolvendo uma classe de funções não-necessariamente lineares, denominadas quasi-isometrias.
In this work we present two theorems proved by Gorak in 2011. These results are generalizations of the Banach-Stone Theorem envolving a class of not-necessarily linear functions, called quasi-isometries.

Para um espaço localmente compacto K e um espaço de Banach X, seja C_0(K,X) o espaço das funções continuas que se anulam no infinito munido da norma do supremo. Nesta tese se provam resultados relacionados com a geometria destes espaços.
For a locally compact Hausdorff space K and a Banach spaces X, let C_0(K,X) be the Banach space of continuous functions which vanish at infinity endowed with the supremum norm. We prove some results about geometry of these spaces.

A classificação isomorfa dos espaços de Banach separáveis C(K) é devida a Milutin no caso em que K são não enumeráveis e a Bessaga e Pelczynski no caso em que K são enumeráveis. Neste trabalho apresentamos uma extensão vetorial dessa classificação e tiramos várias consequências, por exemplo, considerando o espaço métrico compacto infinito K e Y um espaço de Banach:     1. Sendo 1 < p < ∞ e Γ um conjunto infinito, classificamos, a menos de isomorfismo, os espaços de Banach C(K, Y ⊕ lp(Γ)), quando o dual de Y contém uma cópia de lq, onde 1/p+ 1/q =1.     2. Classificamos os espaços de Banach C(K, Y ⊕ l∞(Γ)), quando a densidade de Y é estritamente menor que 2|Γ|.     3. Classificamos os espaços de Banach C(K ×(S⊕ βΓ)) e C(S ⊕ (K× βΓ)), onde S é um compacto disperso de Hausdorff arbitrário e βΓ é a compactificação de Stone-Cech de Γ. Obtemos, também, algumas leis de cancelamento para espaços de Banach da forma C(K1,X)⊕ C(K2,Y), onde K1 e K2 são espaços compactos métricos infinitos de Hausdorff e X, Y espaços de Banach satisfazendo condições adequadas. Estabelecemos também um teorema de quase-dicotomia envolvendo os espaços C(K,X), onde X tem cotipo finito. Finalmente, apresentamos algumas majorações nas distorções de isomorfismos positivos de C([0,ωk]) em C([0,ω]) e também de C([0,ω]) em C([0,ωk]), k∈ N, k ≥ 2.
The isomorphic classification of separable Banach spaces C(K) is due Milutin in the case when K are uncountable and to Bessaga and Pelczynski in the case when K are countable. In this work we prove a vectorial extention of this classification and provide several consequences, for example considering the infinite metric compact space K and Y a Banach space:     1. Let 1 < p < ∞ and Γ a infinite set, we classify, up to an isomorphism, the Banach spaces C(K, Y ⊕ lp(Γ)), in the case where the dual of Y contains no copy of lq, where 1/p+ 1/q =1.     2. We classify the Banach spaces C(K, Y ⊕ l∞(Γ)), when the density character of Y is strictly less that 2|Γ|.     3. We classify the Banach spaces C(K ×(S⊕ βΓ)) and C(S ⊕ (K× βΓ)) where S is an arbitrary dispersed compact and βΓ is the Stone-Cech compactification of Γ. We obtain also some cancellation laws for Banach spaces in the form C(K1,X)⊕ C(K2,Y), where K1 and K2 are metric compact Hausdorff spaces and X, Y Banach spaces satisfying appropriate conditions. We established also a quasi-dichotomy theorem envolving the C(K,X) spaces, where X is of finite cotype. Finally, we present some upper bounds of distortions of positive isomorphisms of C([0,ωk]) on C([0,ω]) and also of C([0,ω]) on C([0,ωk]), k∈ N, k ≥ 2.

We define and study a new family of Banach spaces, the J ames-Schreier spaces, cre- ated by combining key properties in the definitions of two important classical Banach spaces, namely James' quasi-reflexive space and Schreier's space. We explore both the Banach space and Banach algebra theory of these spaces. The new spaces inherit aspects of both parent spaces: our main results are that the J ames-Schreier spaces each have a shrinking basis, do not embed in a Banach space with an unconditional basis, and each of their closed, infinite-dimensional subspaces contains a copy of Co. As Banach sequence algebras each James-Schreier space has a bounded approx- imate identity and is weakly amenable but not amenable, and the bidual and multiplier - algebra are isometrically isomorphic. We approach our study of Banach sequence algebras from the point of view of Schauder basis theory, in particular looking at those Banach sequence algebras for which the unit vectors form an unconditional or shrinking basis. We finally show that for each Banach space X with an unconditional basis we may construct a James-like Banach sequence algebra j(X) with a bounded approximate identity, and give a condition on the shift operators acting on X which implies that j(X) will contain a copy of X as a complemented ideal and hence not be amenable.

Capítulo 1. Después de estudiar algunos preliminares sobre familias adecuadas de conjuntos, formulamos y probamos algunas equivalencias, cada una de ellas son una condición suficiente para que la familia defina un conjunto compacto de Gul'ko. Damos una caracterización de conjunto compacto de Gul'ko en términos de emparejamiento con un conjunto $\mathcal{K}$-analítico. Capítulo 2. Estudiamos propiedades de los espacios de Banach débilmente Lindelöf determinados no-separables. Damos una caracterización por medio de la existencia de un generador proyeccional full sobre él. Estudiamos algunos aspectos sobre sistemas biortogonales en espacios de Banach. Usando técnicas de resoluciones proyeccionales de la identidad, probamos una extensión de un resultado de Argyros y Mercourakis. Capítulo 3. En el espacio $(c_0(\Gamma),\|\cdot\|_\infty)$, con $\Gamma\in\mathbb{R}$, damos una norma equivalente estrictamente convexa. Capítulo 4. Consideramos una caracterización de los subespacios de espacios de Banach débilmente compactamente generados, en términos de una propiedad de cubrimiento de la bola unidad por medio de conjuntos $\epsilon$-débilmente compactos. Reemplazamos este concepto por otro más preciso que llamamos $\epsilon$-débilmente auto-compactos, este concepto permite una mejor descripción. Capítulo 5. Damos condiciones intrínsecas, necesarias y suficientes para que un espacio de Banach sea generado por $c_0(\Gamma)$ o $\ell_p(\Gamma)$ para $p\in(1,+\infty)$. Ofrecemos una nueva demostración de un resultado de Rosenthal, sobre operadores de $c_0(\Gamma)$ en un espacio de Banach.
González Correa, AL. (2008). Compacta in Banach spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8312
Palancia

We study the geometric classification of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as defined by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (Ω,B,??).

The notion of asymptotic structure of an infinite dimensional Banach space was introduced by Maurey, Milman and Tomczak-Jaegermann. The asymptotic structure consists of those finite dimensional spaces which can be found everywhere `at infinity'. These are defined as the spaces for which there is a winning strategy in a certain vector game. The above authors introduced the class of asymptotic $\ell_p$ spaces, which are the spaces having simplest possible asymptotic structure. Key examples of such spaces are Tsirelson's space and James' space. We prove some new properties of general asymptotic $\ell_p$ spaces and also compare the notion of asymptotic $\ell_2$ with other notions of asymptotic Hilbert space behaviour such as weak Hilbert and asymptotically Hilbertian. We study some properties of smooth functions defined on subsets of asymptotic $\ell_\infty$ spaces. Using these results we show that that an asymptotic $\ell_\infty$ space which has a suitably smooth norm is isomorphically polyhedral, and therefore admits an equivalent analytic norm. We give a sufficient condition for a generalized Orlicz space to be a stabilized asymptotic $\ell_\infty$ space, and hence obtain some new examples of asymptotic $\ell_\infty$ spaces. We also show that every generalized Orlicz space which is stabilized asymptotic $\ell_\infty$ is isomorphically polyhedral. In 1991 Gowers and Maurey constructed the first example of a space which did not contain an unconditional basic sequence. In fact their example had a stronger property, namely that it was hereditarily indecomposable. The space they constructed was `$\ell_1$-like' in the sense that for any $n$ successive vectors $x_1 < \ldots < x_n$, $\frac{1}{f(n)} \sum_{i=1}^n \| x_i \| \leq \| \sum_{i=1}^n x_i \| \leq \sum_{i=1}^n \| x_i \|,$ where $ f(n) = \log_2 (n+1) $. We present an adaptation of this construction to obtain, for each $ p \in (1, \infty)$, an hereditarily indecomposable Banach space, which is `$\ell_p$-like' in the sense described above. We give some sufficient conditions on the set of types, $\mathscr{T}(X)$, for a Banach space $X$ to contain almost isometric copies of $\ell_p$ (for some $p \in [1, \infty)$) or of $c_0$. These conditions involve compactness of certain subsets of $\mathscr{T}(X)$ in the strong topology. The proof of these results relies heavily on spreading model techniques. We give two examples of classes of spaces which satisfy these conditions. The first class of examples were introduced by Kalton, and have a structural property known as Property (M). The second class of examples are certain generalized Tsirelson spaces. We introduce the class of stopping time Banach spaces which generalize a space introduced by Rosenthal and first studied by Bang and Odell. We look at subspaces of these spaces which are generated by sequences of independent random variables and we show that they are isomorphic to (generalized) Orlicz spaces. We deduce also that every Orlicz space, $h_\phi$, embeds isomorphically in the stopping time Banach space of Rosenthal. We show also, by using a suitable independence condition, that stopping time Banach spaces also contain subspaces isomorphic to mixtures of Orlicz spaces.

Title from PDF of title page (University of Missouri--Columbia, viewed on Feb. 20, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Professor Nigel J. Kalton. Vita. Includes bibliographical references.

Tensor products of Banach Spaces are studied. An introduction to tensor products is given. Some results concerning the reciprocal Dunford-Pettis Property due to Emmanuele are presented. Pelczyriski's property (V) and (V)-sets are studied. It will be shown that if X and Y are Banach spaces with property (V) and every integral operator from X into Y* is compact, then the (V)-subsets of (X⊗F)* are weak* sequentially compact. This in turn will be used to prove some stronger convergence results for (V)-subsets of C(Ω,X)*.

The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

Our aim in this research was to study monotone operators in Banach spaces. In particular, the most important concept in this theory, the maximal monotone operators. Here we make an attempt to describe most of the important results and concepts on maximal monotone operators and how they all tie together. We will take a brief look at subdifferentials, which generalize the notion of a derivative. The subdifferential is a maximal monotone operator and it has proved to be of fundamental importance for the study of maximal monotone operators. The theory of maximal monotone operators is somewhat complete in reflexive Banach spaces. However, in nonreflexive Banach spaces it is still to be developed fully. As such, here we will describe most of the important results about maximal monotone operators in Banach spaces and we will distinguish between the reflexive Banach spaces and nonreflexive Banach spaces when a property is known to hold only in reflexive Banach spaces. In the latter case, we will state what the corresponding situation is in nonreflexive Banach spaces and we will give counter examples whenever such a result is known to fail in nonreflexive Banach spaces. The representations of monotone operators by convex functions have found to be extremely useful for the study of maximal monotone operators and it has generated a lot of interest of late. We will discuss some of those key representations and their properties. We will also demonstrate how these representations could be utilized to obtain results about maximal monotone operators. We have included a discussion about the very important Rockafellar sum theorem and some its generalizations. This key result and its generalizations have only been proved in reflexive Banach spaces. We will also discuss several special cases where the Rockafellar sum theorem is known to be true in nonreflexive Banach spaces. The subclasses which provide a basis for the study of monotone operators in nonreflexive Banach spaces are also discussed here

We define a universal Fréchet set S of a Banach space Y as a subset containing a point of Fréchet differentiability of every Lipschitz function g : Y -> R. We prove a sufficient condition for S to be a universal Fréchet set and use this to construct new examples of such sets. The strongest such result says that in a non-zero Banach space Y with separable dual one can find a universal Fréchet set S ⊆ Y that is closed, bounded and has Hausdorff dimension one.

The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of $\ell_2$ isometrically. We say that a Banach space has the Finite Isometric Krivine Property (FIKP) if whenever $\ell_p$ is finitely representable in the space then it contains every subset of $\ell_p$ isometrically. We show that every infinite-dimensional Banach space \emph{nearly} has FIDP and every Banach space nearly has FIKP. We then use convexity arguments to demonstrate that not every Banach space has FIKP, and thus we can exhibit classes of Banach spaces that fail to have Property AI-I. The methods used break down when one attempts to prove that there is a Banach space without FIDP and we conjecture that every infinite-dimensional Banach space has Property FIDP.

The aim of this thesis is to provide a survey of the topic of composition operators on spaces of (equivalence classes of) measurable functions and attempt to unify some of the most important results contained in the literature. A large class of these spaces can be equipped with norms turning them into Banach lattices. These spaces are called Banach function spaces and examples include the Lebesgue, Lorentz, Orlicz and Orlicz-Lorentz spaces.

[EN] The Ph.D. thesis "Weighted Banach Spaces of harmonic functions" presented here, treats several topics of functional analysis such as weights, composition operators, Fréchet and Gâteaux differentiability of the norm and isomorphism classes. The work is divided into four chapters that are preceded by one in which we introduce the notation and the well-known properties that we use in the proofs in the rest of the chapters. In the first chapter we study Banach spaces of harmonic functions on open sets of R^d endowed with weighted supremun norms. We define the harmonic associated weight, we explain its properties, we compare it with the holomorphic associated weight introduced by Bierstedt, Bonet and Taskinen, and we find differences and conditions under which they are exactly the same and conditions under which they are equivalent. The second chapter is devoted to the analysis of composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions. We characterize the continuity, the compactness and the essential norm of composition operators among these spaces in terms of their weights, thus extending the results of Bonet, Taskinen, Lindström, Wolf, Contreras, Montes and others for composition operators between spaces of holomorphic functions. We prove that for each value of the interval [0,1] there is a composition operator between weighted spaces of harmonic functions such that its essential norm attains this value. Most of the contents of Chapters 1 and 2 have been published by E. Jordá and the author in [48]. The third chapter is related with the study of Gâteaux and Fréchet differentiability of the norm. The \v{S}mulyan criterion states that the norm of a real Banach space X is Gâteaux differentiable at x\inX if and only if there exists x^* in the unit ball of the dual of X weak^* exposed by x and the norm is Fréchet differentiable at x if and only if x^* is weak^* strongly exposed in the unit ball of the dual of X by x. We show that in this criterion the unit ball of the dual of X can be replaced by a smaller convenient set, and we apply this extended criterion to characterize the points of Gâteaux and Fréchet differentiability of the norm of some spaces of harmonic functions and continuous functions with vector values. Starting from these results we get an easy proof of the theorem about the Gâteaux differentiability of the norm for spaces of compact linear operators announced by Heinrich and published without proof. Moreover, these results allow us to obtain applications to classical Banach spaces as the space H^\infty of bounded holomorphic functions in the disc and the algebra A(\overline{\D}) of continuous functions on \overline{\D} which are holomorphic on \D. The content of this chapter has been included by E. Jordá and the author in [47]. Finally, in the forth chapter we show that for any open set U of R^d and weight v on U, the space hv0(U) of harmonic functions such that multiplied by the weight vanishes at the boundary on U is almost isometric to a closed subspace of c0, extending a theorem due to Bonet and Wolf for the spaces of holomorphic functions Hv0(U) on open sets U of C^d. Likewise, we also study the geometry of these weighted spaces inspired by a work of Boyd and Rueda, examining topics such as the v-boundary and v-peak points and we give the conditions that provide examples where hv0(U) cannot be isometric to c0. For a balanced open set U of R^d, some geometrical conditions in U and convexity in the weight v ensure that hv0(U) is not rotund. These results have been published by E. Jordá and the author [46].
[ES] La presente memoria, "Espacios de Banach ponderados de funciones armónicas ", trata diversos tópicos del análisis funcional, como son las funciones peso, los operadores de composición, la diferenciabilidad Fréchet y Gâteaux de la norma y las clases de isomorfismos. El trabajo está dividido en cuatro capítulos precedidos de uno inicial en el que introducimos la notación y las propiedades conocidas que usamos en las demostraciones del resto de capítulos. En el primer capítulo estudiamos espacios de Banach de funciones armónicas en conjuntos abiertos de R^d dotados de normas del supremo ponderadas. Definimos el peso asociado armónico, explicamos sus propiedades, lo comparamos con el peso asociado holomorfo introducido por Bierstedt, Bonet y Taskinen, y encontramos diferencias y condiciones para que sean exactamente iguales y condiciones para que sean equivalentes. El capítulo segundo está dedicado al análisis de los operadores de composición con símbolo holomorfo entre espacios de Banach ponderados de funciones pluriarmónicas. Caracterizamos la continuidad, la compacidad y la norma esencial de operadores de composición entre estos espacios en términos de los pesos, extendiendo los resultados de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes y otros para operadores de composición entre espacios de funciones holomorfas. Probamos que para todo valor del intervalo [0,1] existe un operador de composición sobre espacios ponderados de funciones armónicas tal que su norma esencial alcanza dicho valor. La mayoría de los contenidos de los capítulos 1 y 2 han sido publicados por E. Jordá y la autora en [48]. El capítulo tercero está relacionado con el estudio de la diferenciabilidad Gâteaux y Fréchet de la norma. El criterio de \v{S}mulyan establece que la norma de un espacio de Banach real X es Gâteaux diferenciable en x\in X si y sólo si existe x^* en la bola unidad del dual de X débil expuesto por x y la norma es Fréchet diferenciable en x si y sólo si x^*es débil fuertemente expuesto en la bola unidad del dual de X por x. Mostramos que en este criterio la bola del dual de X puede ser reemplazada por un conjunto conveniente más pequeño, y aplicamos este criterio extendido para caracterizar los puntos de diferenciabilidad Gâteaux y Fréchet de la norma de algunos espacios de funciones armónicas y continuas con valores vectoriales. A partir de estos resultados conseguimos una prueba sencilla del teorema sobre la diferenciabilidad Gâteaux de la norma de espacios de operadores lineales compactos enunciado por Heinrich y publicado sin la prueba. Además, éstos nos permiten obtener aplicaciones para espacios de Banach clásicos como H^\infty de funciones holomorfas acotadas en el disco y A(\overline{\D}) de funciones continuas en \overline{\D} que son holomorfas en \D. Los contenidos de este capítulo han sido incluidos por E. Jordá y la autora en [47]. Finalmente, en el capítulo cuarto mostramos que para cualquier abierto U contenido en R^d y cualquier peso v en U, el espacio hv0(U), de funciones armónicas tales que multiplicadas por el peso desaparecen en el infinito de U, es casi isométrico a un subespacio cerrado de c0, extendiendo un teorema debido a Bonet y Wolf para los espacios de funciones holomorfas Hv0(U) en abiertos U de C^d. Así mismo, inspirados por un trabajo de Boyd y Rueda también estudiamos la geometría de estos espacios ponderados examinando tópicos como la v-frontera y los puntos v-peak y damos las condiciones que proporcionan ejemplos donde hv0(U) no puede ser isométrico a c0. Para un conjunto abierto equilibrado U de R^d, algunas condiciones geométricas en U y sobre convexidad en el peso v aseguran que hv0(U) no es rotundo. Estos resultados han sido publicados por E. Jordá y la autora en [46].
[CAT] La present memòria, "Espais de Banach ponderats de funcions harmòniques", tracta diversos tòpics de l'anàlisi funcional, com són les funcions pes, els operadors de composició, la diferenciabilitat Fréchet i Gâteaux de la norma i les clases d'isomorfismes. El treball està dividit en quatre capítols precedits d'un d'inicial en què introduïm la notació i les propietats conegudes que fem servir en les demostracions de la resta de capítols. En el primer capítol estudiem espais de Banach de funcions harmòniques en conjunts oberts de R^d dotats de normes del suprem ponderades. Definim el pes associat harmònic, expliquem les seues propietats, el comparem amb el pes associat holomorf introduït per Bierstedt, Bonet i Taskinen, i trobem diferències i condicions perquè siguen exactament iguals i condicions perquè siguen equivalents. El capítol segon està dedicat a l'anàlisi dels operadors de composició amb símbol holomorf entre espais de Banach ponderats de funcions pluriharmòniques. Caracteritzem la continuïtat, la compacitat i la norma essencial d'operadors de composició entre aquests espais en termes dels pesos, estenent els resultats de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes i altres per a operadors de composició entre espais de funcions holomorfes. Provem que per a tot valor de l'interval [0,1] hi ha un operador de composició sobre espais ponderats de funcions harmòniques tal que la seua norma essencial arriba aquest valor. La majoria dels continguts dels capítols 1 i 2 han estat publicats per E. Jordá i l'autora en [48]. El capítol tercer està relacionat amb l'estudi de la diferenciabilitat Gâteaux y Fréchet de la norma. El criteri de \v{S}mulyan estableix que la norma d'un espai de Banach real X és Gâteaux diferenciable en x\inX si i només si existeix x^* a la bola unitat del dual de X feble exposat per x i la norma és Fréchet diferenciable en x si i només si x^* és feble fortament exposat a la bola unitat del dual de X per x. Mostrem que en aquest criteri la bola del dual de X pot ser substituïda per un conjunt convenient més petit, i apliquem aquest criteri estès per caracteritzar els punts de diferenciabilitat Gâteaux i Fréchet de la norma d'alguns espais de funcions harmòniques i contínues amb valors vectorials. A partir d'aquests resultats aconseguim una prova senzilla del teorema sobre la diferenciabilitat Gâteaux de la norma d'espais d'operadors lineals compactes enunciat per Heinrich i publicat sense la prova. A més, aquests ens permeten obtenir aplicacions per a espais de Banach clàssics com l'espai H^\infty de funcions holomorfes acotades en el disc i l'àlgebra A(\overline{\D}) de funcions contínues en \overline{\D} que són holomorfes en \D. Els continguts d'aquest capítol han estat inclosos per E. Jordá i l'autora en [47]. Finalment, en el capítol quart mostrem que per a qualsevol conjunt obert U de R^d i qualsevol pes v en U, l'espai hv0(U), de funcions harmòniques tals que multiplicades pel pes desapareixen en el infinit d'U, és gairebé isomètric a un subespai tancat de c0, estenent un teorema degut a Bonet y Wolf per als espais de funcions holomorfes Hv0(U) en oberts U de C^d. Així mateix, inspirats per un treball de Boyd i Rueda també estudiem la geometria d'aquests espais ponderats examinant tòpics com la v-frontera i els punts v-peak i donem les condicions que proporcionen exemples on hv0(U) no pot ser isomètric a c0. Per a un conjunt obert equilibrat U de R^d, algunes condicions geomètriques en U i sobre convexitat en el pes v asseguren que hv0(U) no és rotund. Aquests resultats han estat publicats per E. Jordá i l'autora en [46].
Zarco García, AM. (2015). Weighted Banach spaces of harmonic functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56461
TESIS

A continuous convex function on an open interval of the real line is differentiable everywhere except on a countable subset of its domain. There has been interest in the problem of characterizing those Banach spaces where the continuous functions exhibit similar differentiability properties. In this paper we show that if a Banach space E has property (H*) and B_E• is weak* sequentially compact, then E is an Asplund space. In the case where the space is weakly compactly generated, it is shown that property (H*) is equivalent for the space to admit an equivalent Frechet differentiable norm. Moreover, we define the SH* spaces, show that every SH* space is an Asplund space, and show that every weakly sequentially complete SH* space is reflexive. Also, we study the relation between property (H*) and the asymptotic norming property (ANP). By a slight modification of the ANP we define the ANP*, and show that if the dual of a Banach spaces has the ANP*-I then the space admits an equivalent Fréchet differentiability norm, and that the ANP*-II is equivalent to the space having property (H*) and the closed unit ball of the dual is weak* sequentially compact. Also, we show that in the dual of a weakly countably determined Banach space all the ANP-K'S are equivalent, and they are equivalent for the predual to have property (H*).

The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

A set S in a Banach space X is called a universal differentiability set if S contains a point of differentiability of every Lipschitz function f : X -> R. The present thesis investigates the nature of such sets. We uncover examples of exceptionally small universal differentiability sets and prove that all universal differentiability sets satisfy certain strong structural conditions. Later, we expand our focus to properties of more general absolutely continuous functions.

Bibliography: leaves 87-90.
Function spaces have been a useful tool in probing the convergence of sequences of functions. The theory seems to have been triggered off by the works of Ascoli [36], Arzelà [37] and Hadamard [38]. In this thesis, we consider the space of continuous functions from a topological space X into the reals R, which we denote C(X).

The main theme of the thesis is the study of continuity and approximation problems, involving matrix-valued and vector-valued Hardy spaces on the unit disc ID and its boundary T in the complex plane. The first part of the thesis looks at the factorization of square matrix-valued boundary functions, beginning with spectral factorization in Chapter 2. Then ideas involving approximations with inner and outer functions are used to solve a matrix analogue of the Douglas-Rudin problem in Chapter 3. In both cases, considerable considerable extra difficulties are created by the noncommutativity of matrix multiplication. More specifically, we show that the matrix spectral factorization mapping is sequentially continuous from LP to H2p (where 1