Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2016 | Buch

Bases and Basic Sequences Kapitel lesen Erstes Kapitel lesen

Topics in Banach Space Theory

verfasst von: Fernando Albiac, Nigel J. Kalton

Verlag: Springer International Publishing

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This text provides the reader with the necessary technical tools and background to reach the frontiers of research without the introduction of too many extraneous concepts. Detailed and accessible proofs are included, as are a variety of exercises and problems. The two new chapters in this second edition are devoted to two topics of much current interest amongst functional analysts: Greedy approximation with respect to bases in Banach spaces and nonlinear geometry of Banach spaces. This new material is intended to present these two directions of research for their intrinsic importance within Banach space theory, and to motivate graduate students interested in learning more about them.

This textbook assumes only a basic knowledge of functional analysis, giving the reader a self-contained overview of the ideas and techniques in the development of modern Banach space theory. Special emphasis is placed on the study of the classical Lebesgue spaces Lp (and their sequence space analogues) and spaces of continuous functions. The authors also stress the use of bases and basic sequences techniques as a tool for understanding the isomorphic structure of Banach spaces.

From the reviews of the First Edition:

"The authors of the book…succeeded admirably in creating a very helpful text, which contains essential topics with optimal proofs, while being reader friendly… It is also written in a lively manner, and its involved mathematical proofs are elucidated and illustrated by motivations, explanations and occasional historical comments… I strongly recommend to every graduate student who wants to get acquainted with this exciting part of functional analysis the instructive and pleasant reading of this book…"—Gilles Godefroy, Mathematical Reviews

Inhaltsverzeichnis

Frontmatter
Chapter 1. Bases and Basic Sequences
Abstract
In this chapter we are going to introduce the fundamental notion of a Schauder basis of a Banach space and the corresponding notion of a basic sequence. One of the key ideas in the isomorphic theory of Banach spaces is to use the properties of bases and basic sequences as a tool to understanding the differences and similarities between spaces. The systematic use of basic sequence arguments also turns out to simplify some classical theorems, and we illustrate this with the Eberlein–S̆mulian theorem on weakly compact subsets of a Banach space.
Fernando Albiac, Nigel J. Kalton
Chapter 2. The Classical Sequence Spaces
Abstract
We now turn to the classical sequence spaces ℓ p for \(1 \leq p <\infty\) and c 0 . The techniques developed in the previous chapter will prove very useful in this context. These Banach spaces are, in a sense, the simplest of all Banach spaces, and their structure has been well understood for many years. However, if p ≠ 2, there can still be surprises, and there remain intriguing open questions.
Fernando Albiac, Nigel J. Kalton
Chapter 3. Special Types of Bases
Abstract
Knowing whether a separable Banach space has a Schauder basis and identifying one that allows one to compute easily the norm of its elements is important.
Fernando Albiac, Nigel J. Kalton
Chapter 4. Banach Spaces of Continuous Functions
Abstract
We are now going to shift our attention from sequence spaces to spaces of functions, and we start in this chapter by considering spaces of type \(\mathcal{C}(K)\).
Fernando Albiac, Nigel J. Kalton
Chapter 5. L 1(μ)-Spaces and ??(K)-Spaces
Abstract
In this chapter we will prove some very classical results concerning weak compactness and weakly compact operators on \(\mathcal{C}(K)\)-spaces and L 1(μ)-spaces, and exploit them to give further information about complemented subspaces of such spaces.
Fernando Albiac, Nigel J. Kalton
Chapter 6. The Spaces L p for 1 ≤ p < ∞
Abstract
In this chapter we will initiate the study of the Banach space structure of the spaces L p (μ), where \(1 \leq p <\infty.\) We will be interested in some natural questions that ask which Banach spaces can be isomorphic to a subspace of a space L p (μ). Questions of this type were called problems of linear dimension by Banach in his book [18].
Fernando Albiac, Nigel J. Kalton
Chapter 7. Factorization Theory
Abstract
This chapter is devoted to some important results on factorization of operators. Suppose X, Y are Banach spaces and that T: X → Y is a continuous operator. We say that T factorizes through a Banach space E if there are continuous operators R: X → E and S: E → Y such that T = SR. Pictorially, we have the following diagram:
Fernando Albiac, Nigel J. Kalton
Chapter 8. Absolutely Summing Operators
Abstract
The theory of absolutely summing operators was one of the most profound developments in Banach space theory between 1950 and 1970. It originates in a fundamental paper of Grothendieck [121] (which actually appeared in 1956). However, some time passed before Grothendieck’s remarkable work really became well known among specialists. There are several reasons for this. One major point is that Grothendieck stopped working in the field at just about this time and moved into algebraic geometry (his work in algebraic geometry earned the Fields Medal in 1966). Thus he played no role in the dissemination of his own ideas. He also chose to publish in a relatively obscure journal that was not widely circulated; before the advent of the Internet it was much more difficult to track down copies of articles. Thus it was not until the 1968 paper of Lindenstrauss and Pełczyński [196] that Grothendieck’s ideas became widely known. Since 1968, the theory of absolutely summing operators has become a cornerstone of modern Banach space theory.
Fernando Albiac, Nigel J. Kalton
Chapter 9. Perfectly Homogeneous Bases and Their Applications
Abstract
In this chapter we first prove a characterization of the canonical bases of the spaces ℓ p (1 ≤ p < ∞) and c 0 due to Zippin [308]. In the remainder of the chapter we show how this is used in several different contexts to prove general theorems by reduction to the ℓ p case. For example, we show that the Lindenstrauss–Pełczyński theorem on the uniqueness of the unconditional basis in c 0, ℓ 1, and ℓ 2 (Theorem 8.​3.​3) has a converse due to Lindenstrauss and Zippin; these are the only three such spaces. We also deduce a characterization of c 0 and ℓ p in terms of complementation of block basic sequences due to Lindenstrauss and Tzafriri [200] and apply it to prove a result of Pełczyński and Singer [249] on the existence of conditional bases in any Banach space with a basis.
Fernando Albiac, Nigel J. Kalton
Chapter 10. Greedy-Type Bases
Abstract
Suppose that X is a Banach space and that \(\mathcal{B} = (e_{n})_{n=1}^{\infty }\) is a basis of X. An m-term approximation with respect to \(\mathcal{B}\) is a map T m : X → X such that for each x ∈ X, T m (x) is a linear combination of at most m elements of \(\mathcal{B}\). An approximation algorithm is a sequence (T m ) m = 1 ∞ of such maps.
Fernando Albiac, Nigel J. Kalton
Chapter 11. ℓ p -Subspaces of Banach Spaces
Abstract
In the previous chapters the spaces ℓ p (1 ≤ p < ∞) and c 0 played a pivotal role in the development of the theory. This suggests that we should ask when we can embed one of these spaces in an arbitrary Banach space. For c 0 we have a complete answer: c 0 embeds into X if and only if X contains a WUC series that is not unconditionally convergent (Theorem 2.​4.​11).
Fernando Albiac, Nigel J. Kalton
Chapter 12. Finite Representability of ℓ p -Spaces
Abstract
We are now going to switch gears and study local properties of infinite-dimensional Banach spaces. In Banach space theory the word local is used to denote finite-dimensional. We can distinguish between properties of a Banach space that are determined by its finite-dimensional subspaces and properties that require understanding of the whole space. For example, one cannot decide that a space is reflexive just by looking at its finite-dimensional subspaces, but properties like type and cotype that depend on inequalities with only finitely many vectors are local in character.
Fernando Albiac, Nigel J. Kalton
Chapter 13. An Introduction to Local Theory
Abstract
The aim of this chapter is to provide an introduction to the ideas of the local theory and a quantitative proof of Dvoretzky’s theorem.
Fernando Albiac, Nigel J. Kalton
Chapter 14. Nonlinear Geometry of Banach Spaces
Abstract
A Banach space is, by its nature, also a metric space. When we identify a Banach space with its underlying metric space, we choose to forget its linear structure. The fundamental question of nonlinear geometry is to determine to what extent the metric structure of a Banach space already determines its linear structure. What can be said of two Banach spaces that are Lipschitz-isomorphic? uniformly homeomorphic? or coarsely Lipschitz-isomorphic in the spirit of M. Gromov’s geometric theory of groups? These questions are still partly open, and investigating them requires some quite technical tools. In this chapter, we will consider only basic techniques whose purpose is to produce specific linear maps from nonlinear ones (typically, Lipschitz maps).
Fernando Albiac, Nigel J. Kalton
Chapter 15. Important Examples of Banach Spaces
Abstract
In this last, optional chapter, we construct some examples of Banach spaces that played an important role in the development of Banach space theory. These constructions are not elementary, so we have preferred to remove them from the main text.
Fernando Albiac, Nigel J. Kalton
Backmatter
Metadaten
Titel
Topics in Banach Space Theory
verfasst von
Fernando Albiac
Nigel J. Kalton
Copyright-Jahr
2016
Electronic ISBN
978-3-319-31557-7
Print ISBN
978-3-319-31555-3
DOI
https://doi.org/10.1007/978-3-319-31557-7