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Relevant bibliographies by topics / Smoothness and renorming

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Academic literature on the topic 'Smoothness and renorming'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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Journal articles on the topic "Smoothness and renorming"

1

Finet, Catherine. "Renorming Banach spaces with many projections and smoothness properties." Mathematische Annalen 284, no. 4 (December 1989): 675–79. http://dx.doi.org/10.1007/bf01443358.

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García-Pacheco, Francisco Javier. "Renormings concerning exposed points and non-smoothness." Science in China Series A: Mathematics 52, no. 9 (September 2009): 1844–48. http://dx.doi.org/10.1007/s11425-009-0155-y.

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Diestel, J. "Book Review: Smoothness and renormings in Banach spaces." Bulletin of the American Mathematical Society 31, no. 1 (July 1, 1994): 140–42. http://dx.doi.org/10.1090/s0273-0979-1994-00500-2.

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Causey, R., A. Fovelle, and G. Lancien. "Asymptotic smoothness in Banach spaces, three-space properties and applications." Transactions of the American Mathematical Society, December 16, 2022. http://dx.doi.org/10.1090/tran/8818.

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We study four asymptotic smoothness properties of Banach spaces, denoted T p , A p , N p \mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p , and P p \mathsf {P}_p . We complete their description by proving the missing renorming characterization for A p \mathsf {A}_p . We show that asymptotic uniform flattenability (property T ∞ \mathsf {T}_\infty ) and summable Szlenk index (property A ∞ \mathsf {A}_\infty ) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of A p \mathsf {A}_p and N p \mathsf {N}_p in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.

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Wójcik, Paweł. "When are maps preserving semi-inner products linear?" Aequationes mathematicae, July 3, 2021. http://dx.doi.org/10.1007/s00010-021-00829-3.

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AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

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Dissertations / Theses on the topic "Smoothness and renorming"

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RUSSO, TOMMASO. "ON SOME OPEN PROBLEMS IN BANACH SPACE THEORY." Doctoral thesis, Università degli Studi di Milano, 2019. http://hdl.handle.net/2434/609289.

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The main line of investigation of the present work is the study of some aspects in the analysis of the structure of the unit ball of (infinite-dimensional) Banach spaces. In particular, we analyse some questions concerning the existence of suitable renormings that allow the new unit ball to possess a specific geometric property. The main part of the thesis is, however, dedicated to results of isometric nature, in which the original norm is the one under consideration. One of the main sources for the selection of the topics of investigation has been the recent monograph [GMZ16], entirely dedicated to collecting several open problems in Banach space theory and formulating new lines of investigation. We take this opportunity to acknowledge the authors for their effort, that offered such useful text to the mathematical community. The results to be discussed in our work actually succeed in solving a few of the problems presented in the monograph and are based on the papers [HáRu17, HKR18, HáRu19, HKR••]. Let us say now a few words on how the material is organised. The thesis is divided in four chapters (some whose contents are outlined below) which are essentially independent and can be read in whatsoever order. The unique chapter which is not completely independent from the others is Chapter 4, where we use some results from Chapter 2 and which is, in a sense, the non-separable prosecution of Chapter 3. However, cross-references are few (never implicit) and usually restricted to quoting some result; it should therefore be no problem to start reading from Chapter 4. The single chapters all share the same arrangement. A first section is dedicated to an introduction to the subject of the chapter; occasionally, we also present the proof of known results, in most cases as an illustration of an important technique in the area. In these introductions we strove to be as self-contained as possible in order to help the novel reader to enter the field; consequently, experts in the area may find them somewhat redundant and prefer to skip most parts of them. The first section of each chapter concludes with the statement of our most significant results and a comparison with the literature. The proofs of these results, together with additional results or generalisations, are presented in the remaining sections of the chapter. These sections usually follow closely the corresponding articles (carefully referenced) where the results were presented.

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2

Guirao, Sánchez Antonio José. "Renormamiento en espacios de Banach." Doctoral thesis, Universidad de Murcia, 2007. http://hdl.handle.net/10803/10965.

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La Tesis está compuesta por un capítulo introductorio y cuatro capítulosque pasamos a describir.El Capítulo 2 contiene un análisis de las funciones que son posiblementemódulo de convexidad (m.c.) para un espacio de Banach uniformementeconvexo (UC). Se muestra que las funciones m.c. están caracterizadas,salvo equivalencia, por ciertas propiedades clásicas de éstas.En el Capítulo 3, se estudia la noción de m.c. de una función convexadefinida en un espacio de Banach. Éste es el primer trabajo con resultadosgenerales y completos en espacios de Banach. Se muestra que un espacio essuperreflexivo sii admite una función (UC) definida en todo el espacio.En el Capítulo 4 se resuelve un problema establecido por Godefroy yZizler; un espacio de Banach superreflexivo con base de Schauder admiteuna norma (UC) que hace monótona a la base. Se obtienen mejoras deestimaciones de James y Gurari.En el Capítulo 5 el autor estudia la noción del módulo de cuadratura. Éstepermite reconocer la (UC) y la suavidad uniforme. El autor define laversión local, y prueba varias caracterizaciones del comportamientopuntual de la norma.
The thesis consists of one introductory chapter and four chapterscontaining original mathematical results. Let us pass to a briefdescription of the main results.Chapter 2 contains an analysis of the possible modulus of rotundityfunctions (m.r.f) for a given uniformly rotund (UC) Banach space. It isshown that m.r.f. are characterized, up to equivalence, by certainclassical properties of them.In Chapter 3, the notion of m.r. for a convex function defined on a Banachspace is studied. This seems to be the first instance of rather completegeneral results on Banach spaces. It is shown that a Banach space issuperreflexive iff it admits a (UC) function defined on the whole space.In Chapter 4 a problem asked by Godefroy and Zizler is solved; asuperreflexive Banach space with Schauder basis can be renormed by (UC)norm which makes the given basis monotone. An improvement of a result ofGurarii is an immediate corollary.In Chapter 5 the author studies the notion of modulus of squareness. Itallows to recognize (UC) and uniform smoothness. The author succeeds todefine the local version, and proves various characterizations ofpointwise behaviour of the norm.

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Books on the topic "Smoothness and renorming"

1

Deville, R. Smoothness and renormings in Banach spaces. Harlow, Essex, England: Longman Scientific & Technical, 1993.

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2

Deville, Robert. Smoothness and renormings in Banach spaces. Harlow: Longman Scientific and Technical, 1993.

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3

Zizler, Gilles Godefroy, and Robert Deville. Smoothness and Renormings in Banach Spaces. Wiley & Sons, Incorporated, John, 1993.

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Book chapters on the topic "Smoothness and renorming"

1

Guirao, Antonio José, Vicente Montesinos, and Václav Zizler. "Higher-order smoothness." In Renormings in Banach Spaces, 337–47. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-08655-7_27.

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Guirao, Antonio José, Vicente Montesinos, and Václav Zizler. "Examples on C1-smoothness." In Renormings in Banach Spaces, 279–83. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-08655-7_19.

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3

Lindenstrauss, Joram, David Preiss, and Tiˇser Jaroslav. "Smoothness and Asymptotic Smoothness." In Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179). Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691153551.003.0008.

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This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

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Lindenstrauss, Joram, David Preiss, and Tiˇser Jaroslav. "Smoothness, Convexity, Porosity, and Separable Determination." In Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179). Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691153551.003.0003.

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This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

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