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In this chapter we introduce basic notions and concepts in Banach space theory. As a rule we will work with real scalars, only in a few instances, e.g., in spectral theory, we will use complex scalars. [ABSTRACT FROM AUTHOR]

An indispensable tool in the study of deeper structural properties of a Banach space X is its weak topology, i.e., the topology on X of the pointwise convergence on elements of the dual space X*, or the weak*topology on X*, i.e., the topology on X* of the pointwise convergence on elements of X. The topology on X* of the uniform convergence on the family of all convex balanced and weakly compact subsets of X plays also an important role. All those topologies can be efficiently studied in the general framework of topological vector spaces. [ABSTRACT FROM AUTHOR]

The Hahn-Banach theorem, in the geometrical form, states that a closed and convex set can be separated from any external point by means of a hyperplane. This intuitively appealing principle underlines the role of convexity in the theory. It is the first, and most important, of the fundamental principles of functional analysis. The rich duality theory of Banach spaces is one of its direct consequences. The second fundamental principle, the Banach open mapping theorem, is studied in the rest of the chapter. [ABSTRACT FROM AUTHOR]

In this chapter we shall introduce Schauder bases, an important concept in Banach space theory. Elements of a Banach space with a Schauder basis may be represented as infinite sequences of ˵coordinates,″ which is very natural and useful for analytical work. Although not every separable Banach space admits a Schauder basis (this is a deep and difficult result of Enflo), basic sequences exist in every infinite-dimensional Banach space (Mazur) and are ideal for the study of linear subspaces and quotients. In many respects, the assumption of having a Schauder basis is not very restrictive, and in fact, for naturally defined separable Banach spaces, it is usually easy to find their Schauder basis. This notion has proved to be an extremely useful tool in the study of the structure of classical as well as abstract Banach spaces. [ABSTRACT FROM AUTHOR]

The interplay between the structure of an infinite-dimensional Banach space and properties of its finite-dimensional subspaces belongs to the subject of the local theory of Banach spaces. It is a vast and deep part of Banach space theory intimately related to probability and combinatorics. Our goal is to familiarize the reader with some of its basic notions and results that are accessible without the use of deep probabilistic tools. [ABSTRACT FROM AUTHOR]

The focus of the study in the present chapter is on Banach spaces containing subspaces isomorphic to c0 or ℓ1. We prove Sobczyk΄s theorem on complementability of c0 in separable overspaces, lifting property of ℓ1 and Pełczyński΄s characterization of separable Banach spaces containing ℓ1. We present Rosenthal΄s ℓ1 theorem, Odell–Rosenthal theorem and the Rosenthal–Bourgain–Fremlin–Talagrand theory of Baire-1 functions on Polish spaces. [ABSTRACT FROM AUTHOR]

In this chapter we study superreflexive Banach spaces. These spaces admit many characterizations by means of equivalent renormings, local properties, uniform smoothness, and dentability properties. We also discuss the structure of these spaces and basic sequences in them. [ABSTRACT FROM AUTHOR]

In this chapter we discuss some basic techniques in nonlinear analysis on Banach spaces that are frequently used in applications in related fields. The classical approach uses differentiability, and we discuss this concept in infinite-dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

In this chapter we study separable Asplund spaces, i.e., Banach spaces with a separable dual space. These spaces admit many equivalent characterizations, in particular by means of C1-smooth renormings and differentiability properties of convex functions. Asplund spaces also play an important role in applications. We study basic results in smooth approximation and ranges of smooth nonlinear operators. [ABSTRACT FROM AUTHOR]

In this chapter, we will first discuss the properties of smoothness in ℓp spaces and in Hilbert spaces. Then we study spaces that have countable James boundary in connection with their higher order smoothness, and its applications. In particular, we study spaces of continuous functions on countable compact spaces. [ABSTRACT FROM AUTHOR]

In this chapter, we begin by proving the Brouwer and the Schauder fixed-point theorems. Then we turn to results on homeomorphisms of convex sets and spaces. We prove Keller΄s theorem on homeomorphism of infinite-dimensional compact convex sets in Banach spaces to ]> . We also prove the Kadec theorem on the homeomorphism of every separable reflexive space to a Hilbert space. Then we prove some results on uniform, in particular Lipschitz, homeomorphisms. [ABSTRACT FROM AUTHOR]

The main topic of the present chapter is the dentability of bounded sets and the closely related Radon–Nikodým property (RNP) of Banach spaces. This property has several equivalent characterizations and applications. In particular, Asplund spaces are characterized by the Radon–Nikodým property of their dual spaces. As another application, we show that Lipschitz mappings from separable Banach spaces into Banach spaces with RNP are at some points Gâteaux differentiable. [ABSTRACT FROM AUTHOR]

In this chapter we study weakly compact operators and the related class of Banach spaces that are generated by weakly compact sets (i.e., weakly compactly generated spaces, in short WCG spaces). We focus on their decomposition properties, renormings, and on the topological properties of their dual spaces. We prove that WCG spaces are generated by reflexive spaces. Then we study absolutely summing operator!absolutely summing operators and the Dunford–Pettis property. [ABSTRACT FROM AUTHOR]

This chapter is an introduction to the topological theory of tensor products of Banach spaces. The focus lies on the applications of tensors in the duality theory for spaces of operators, and their structure as Banach spaces. We discuss the role of the approximation property and Enflo΄s example of a Banach space without the approximation property. [ABSTRACT FROM AUTHOR]

In this chapter we study the weak and weak* topologies of Banach spaces in more detail. We discuss several types of compacta (Eberlein, uniform Eberlein, scattered, Corson, and more), weakly Lindelöf determined spaces and properties of tightness countable tightness!weak topology in weak topologies. We discuss some applications in the structural properties of some Banach spaces. [ABSTRACT FROM AUTHOR]

In this chapter we study basic properties of compact operators on Banach spaces. We present the elementary spectral theory of compact operators in Banach spaces, including the spectral radius and properties of eigenvalues. Then we discus basic spectral properties of selfadjoint operators on Hilbert spaces, their spectral decomposition, and show some of the applications of these topics. [ABSTRACT FROM AUTHOR]

In this short chapter we collect, for the reader΄s convenience, some basic definitions and results that are used in the book. A list of sources for them is provided at the end of each section here. [ABSTRACT FROM AUTHOR]

The paper presents basic notions of fuzzy inference systems based on the Takagi-Sugeno fuzzy model. On the basis of this fuzzy inference system and IF-sets introduced by K.T. Atanassov, novel IF-inference systems can be designed. Thus, an IF-inference system is developed for time series prediction. In the next part of the paper we describe ozone prediction by IF-inference systems and the analysis of the results. [ABSTRACT FROM AUTHOR]

The paper presents the design of municipal creditworthiness parameters. Further, a model is designed based on Learning Vector Quantization neural networks for municipal creditworthiness classification. The model is composed of Kohonen΄s Self-organizing Feature Maps (unsupervised learning) whose outputs represent the input of the Learning Vector Quantization neural networks (supervised learning). [ABSTRACT FROM AUTHOR]

The main theme of this chapter is the existence of biorthogonal systems in general nonseparable Banach spaces. An important role is played by the notion of long Schauder bases; the first section introduces this notion, which is a natural generalization of the usual Schauder basis. The first section also contains Plichko's improvement of the natural "exhaustion" argument that yields the existence of a bounded total biorthogonal system in every Banach space. The second section presents Plichko's characterization of spaces with a fundamental biorthogonal system as those spaces that admit a quotient of the same density with a long Schauder basis. In general, a total biorthogonal system as constructed in the first section has a cardinality that corresponds to the w*-density of the dual space. Thus, such a system may be countable for certain nonseparable spaces (most notably all subspaces of $$\ell _\infty$$). It is therefore a priori unclear if every nonseparable subspace of $$\ell _\infty$$ contains an uncountable biorthogonal system. The third section singles out some natural classes of spaces that are obtained "constructively" (representable spaces)— and hence are well-behaved in this respect, as shown by results of Godefroy and Talagrand. However, the general question of the existence of uncountable biorthogonal systems in every nonseparable space is undecidable in ZFC. [ABSTRACT FROM AUTHOR]

In this chapter, we discuss some further applications of techniques of orthogonalization in the geometry of Banach spaces. In particular, this chapter begins by showing the connection between a form of one-sided biorthogonal systems and the existence of support sets in nonseparable Banach spaces. Set theory once again plays an important role because, building on some previous work of Rolewicz, Borwein, Kutzarova, Lazar, Bell, Ginsburg, Todorčević, and others, recent results of Todorčević and Koszmider demonstrate that the existence of support sets in every nonseparable Banach space is undecidable in ZFC. The second section highlights some work of Granero, Jiménez-Sevilla, Moreno, Montesinos, Plichko, and others on the study of nonseparable Banach spaces that do not admit uncountable biorthogonal systems (the existence of such spaces relies on the use of additional axioms such as ♣); these results include characterizations of spaces in which every dual ball is weak*-separable, as well as an improvement of Sersouri's result showing that such spaces must only contain countable ω-independent families. The latter part of the chapter presents several applications of various types of biorthogonal systems. In particular, it is shown that fundamental biorthogonal systems (and even weaker systems) have applications in the study of norm-attaining operators as originated by Lindenstrauss. The attention then shifts to the Mazur intersection property, where, among other things, an application of biorthogonal systems to renorming Banach spaces with the Mazur intersection property is presented. The chapter concludes by showing that every Banach space can be renormed to have only trivial isometries; the proof of this relies on the fact that every Banach space has a total biorthogonal system. [ABSTRACT FROM AUTHOR]