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2018 | Buch

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A Course in Functional Analysis and Measure Theory

verfasst von: Prof. Vladimir Kadets

Verlag: Springer International Publishing

Buchreihe : Universitext

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

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Über dieses Buch

Written by an expert on the topic and experienced lecturer, this textbook provides an elegant, self-contained introduction to functional analysis, including several advanced topics and applications to harmonic analysis.

Starting from basic topics before proceeding to more advanced material, the book covers measure and integration theory, classical Banach and Hilbert space theory, spectral theory for bounded operators, fixed point theory, Schauder bases, the Riesz-Thorin interpolation theorem for operators, as well as topics in duality and convexity theory.

Aimed at advanced undergraduate and graduate students, this book is suitable for both introductory and more advanced courses in functional analysis. Including over 1500 exercises of varying difficulty and various motivational and historical remarks, the book can be used for self-study and alongside lecture courses.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Metric and Topological Spaces

Abstract

Topological, and especially metric spaces, are frequently mentioned and used in courses on mathematical analysis, linear algebra (in which one of the most important examples of metric spaces, the finite-dimensional Euclidean space, is examined), differential geometry (in which geodesic curves and the intrinsic metric of a surface are studied), and also, it goes without saying, in topology courses. For this reason, we only briefly recall the well-known definitions and facts, discuss the terminology and notation adopted in this book, while dwelling in more detail on issues that possibly are not treated in other courses.

Vladimir Kadets

Chapter 2. Measure Theory

Abstract

The reader has undoubtedly already encountered the notion of measure, though perhaps under a different name. For instance, the area of a plane figure, the length of a rectifiable curve, the volume, the mass, are all examples of measures. The chapter covers the following material: algebras and sigma-algebras of sets; Borel sets; finite and countable additivity of measures; atoms, atomic and non-atomic measures; extension of measures; the Lebesgue measure on the interval and on the real line; Lebesgue’s theorem on the differentiability of monotone functions.

Vladimir Kadets

Chapter 3. Measurable Functions

Abstract

Let $({\Omega _1 ,\Sigma _1})$ and $({\Omega _2 ,\Sigma _2})$ be sets endowed with $\sigma $-algebras of subsets. A function $f:\Omega _1 \rightarrow \Omega _2$ is said to be measurable if $f^{- 1} (A) \in \Sigma _1$ for all $A \in \Sigma _2$. Measurable functions play in measure theory the same role that continuous functions do in the theory of topological spaces. In this chapter we present the basic properties of measurable functions, the Lebesgue’s approximation of measurable functions by simple ones, and the relationship between various types of convergence of sequences of measurable functions.

Vladimir Kadets

Chapter 4. The Lebesgue Integral

Abstract

The Lebesgue integral extends the Riemann integral to wider classes of functions. The definition of Lebesgue integral is a little bit more complicated, but the properties are much more convenient in applications. That is why the modern mathematics mainly uses the Lebesgue integral. To strengthen the analogy with the Riemann integral, in this book the theory of the Lebesgue integral is treated on the basis of Fréchet’s definition, namely, by means of the convergent integral sums analogous to the Riemann integral sums. One of the advantages of this approach is the simplicity with which this definition extends to vector-valued functions.

Vladimir Kadets

Chapter 5. Linear Spaces, Linear Functionals, and the Hahn–Banach Theorem

Abstract

The Hahn–Banach theorem on the extension of linear functionals that will be proved in the present chapter (alternatively known as the analytic form of the Hahn–Banach theorem) is one of the most important theorems in functional analysis. It is frequently used, both in the subject itself and in applications of functional analysis to a wide circle of related fields. Some of these applications will be treated in this book. The Hahn–Banach theorem is traditionally regarded as one of the “fundamental principles of functional analysis”. Such “fundamental principles” also include the geometric form of the Hahn–Banach theorem (Chapter 9), Banach’s inverse operator theorem, the open mapping and the closed graph theorems, as well as the Banach–Steinhaus theorem (Chapter 10).

Vladimir Kadets

Chapter 6. Normed Spaces

Abstract

In this chapter we introduce the main objects of functional analysis: normed spaces, Banach spaces, and continuous linear operators. We provide with basic definitions, examples, and some results like the completeness criterion in terms of absolutely convergent series, the continuity criterion for linear operators, the completeness of the dual space, and the possibility to extend a continuous linear operator to the closure of its domain.

Vladimir Kadets

Chapter 7. Absolute Continuity of Measures and Functions. The Connection Between Derivative and Integral

Abstract

In this chapter, after providing with necessary preliminaries, we present Hahn’s theorem on the positivity and negativity sets of a charge, the Radon–Nikodým theorem on absolutely continuous measures and give a complete description of functions $f:[a, b] \rightarrow \mathbb R$ for which on every $[\alpha , \beta ] \subset [a, b]$ the Newton–Leibniz formula

$$ \int _\alpha ^\beta {f'(t)dt} = f(\beta ) - f(\alpha ) $$

holds in the sense of Lebesgue’s integration.

Vladimir Kadets

Chapter 8. The Integral on C(K)

Abstract

In the first three sections of this chapter we treat in detail the integration theory for functions on a compact topological space K. In the last section the results obtained will be applied to the proof of the Riesz–Markov–Kakutani theorem on the general form of linear functionals on the space C(K).

Vladimir Kadets

Chapter 9. Continuous Linear Functionals

Abstract

In this chapter we address the following subjects: the connection between real and complex functionals; the Hahn–Banach extension theorem for continuous functionals; supporting functionals; the annihilator of a subspace; complete systems of elements of a normed space; the Hahn–Banach separation theorem for convex sets; the connection between properties of an operator and those of its adjoint; and the duality between subspaces and quotient spaces.

Vladimir Kadets

Chapter 10. Classical Theorems on Continuous Operators

Abstract

In this chapter we present the open mapping theorem, the inverse operator theorem, the closed graph theorem, and the uniform boundedness principle. All these results belong to the circle of classical “fundamental principles of functional analysis” and have multiple applications. Some of such applications are given in this chapter, in particular applications to complementability of subspaces, to boundedness of partial sums operators with respect to a Schauder basis, and to Fourier series on an interval.

Vladimir Kadets

Chapter 11. Elements of Spectral Theory of Operators. Compact Operators

Abstract

In this chapter we address the following subjects: the spectrum and eigenvalues of an operator; the resolvent and non-emptyness of the spectrum; finite-rank operators, the approximation property and compactness in Banach spaces; compactness criteria for sets in specific spaces; definition and properties of compact operators; operators of the form $I - T$ with T a compact operator; the structure of the spectrum of a compact operator.

Vladimir Kadets

Chapter 12. Hilbert Spaces

Abstract

Among the infinite-dimensional Banach spaces, Hilbert spaces are distinguished by their relative simplicity. In Hilbert spaces we are able to use our geometric intuition to its fullest potential: measuring angles between vectors, applying Pythagoras’ theorem, and using orthogonal projections. Here we do not run into anomalous phenomena such as non-complemented subspaces or, say, linear functionals that do not attain their upper bound on the unit sphere. All separable infinite-dimensional Hilbert spaces are isomorphic to one another. Thanks to this relative simplicity, Hilbert spaces are often used in applications. In fact, whenever possible (true, this is not always the case), one seeks to use the language of Hilbert spaces rather than that of general Banach or topological vector spaces. The theory of operators in Hilbert spaces is developed in much more depth than that in the general case, which is yet another reason why this technique is frequently employed in applications.

Vladimir Kadets

Chapter 13. Functions of an Operator

Abstract

The apparatus of functions of an operator was created to enable the free use of analogies between formulas involving numbers and formulas that involve operators. In this chapter we build this apparatus starting with polynomials in an operator, then extending the definition to continuous functions and finally to bounded Borel-measurable function of a self-adjoint operator. After giving a brief introduction into integration with respect to a vector measure, we define the spectral measure of a self-adjoint operator and demonstrate the representation of functions of an operator as integrals with respect to the spectral measure. Some applications are given.

Vladimir Kadets

Chapter 14. Operators in

Abstract

In this chapter we tell some basic facts about functionals and operators in spaces $L_p$ and apply them to Fourier series in $L_p[0, 2\pi ]$ and Fourier transform in $L_p(-\infty , \infty )$. The contents of the chapter: the Hölder inequality; connections between the spaces $L_p$ for different values of p; weighted integration functionals; the general form of linear functionals on $L_p$; $\delta $-sequences and the Dini theorem; the Fourier transform in $L_1(-\infty , \infty )$; inversion formulas; the Fourier transform and differentiation; the Fourier transform in $L_2(-\infty , \infty )$; the Hadamard three-lines theorem; the Riesz–Thorin theorem; applications to Fourier series and the Fourier transform.

Vladimir Kadets

Chapter 15. Fixed Point Theorems and Applications

Abstract

An element $x \in X$ is called a fixed point of the mapping $f:X \rightarrow X$ if $f(x) = x$. Many problems, looking rather dissimilar at a first glance, from various domains of mathematics, can be reduced to the search for fixed points of appropriate mappings. For this reason each of the theorems on existence of fixed points discussed in the present chapter has numerous and often very elegant applications. Below we present three classical fix-point theorems: the Banach’s theorem on contractive mappings, the Schauder’s principle (which are applied for demonstrations of the Picard and Peano theorems on the existence of a solution to the Cauchy problem for differential equations and to the Lomonosov invariant subspace theorem), and the Kakutani’s theorem about common fixed points of a family of isometries, with application to the existence of a Haar measure on a compact topological group.

Vladimir Kadets

Chapter 16. Topological Vector Spaces

Abstract

There are natural types of convergence on linear spaces of functions with the feature that the convergence cannot be described as convergence with respect to a norm. These are, for instance, pointwise convergence and convergence in measure. Such types of convergence will, with rare exceptions, be the weak and weak$^*$ convergence in Banach spaces — the main objects of study in Chapter 17. An adequate language for describing such convergences is that of topological vector spaces. After giving the necessary preliminaries about filters and ultrafilters on topological spaces, in the present chapter we give axiomatics and terminology of topological vector spaces, speak about boundedness, precompactness and compactness in such spaces, discuss the extensions of the Hahn–Banach theorem, and present the elegant Eidelheit’s interpolation theorem with applications to interpolation by infinitely smooth functions and analytic functions.

Vladimir Kadets

Chapter 17. Elements of Duality Theory

Abstract

In this chapter we mainly deal with relationship between a space (locally convex or Banach) and its dual space. The contents of the chapter: the general notion of duality; weak topology; polars and bipolars; the adjoint operator; Alaoglu’s theorem; $w^*$-convergence in a dual Banach space; the second dual space; weak convergence criteria in classical Banach spaces; total and norming sets of functionals; metrizability conditions; the Eberlein–Smulian theorem; reflexive Banach spaces.

Vladimir Kadets

Chapter 18. The Krein–Milman Theorem and Its Applications

Abstract

One of the main merits of the functional analysis-based approach to problems of classical analysis is that it reduces problems formulated analytically to problems of a geometric character. The geometric objects that arise in this way lie in infinite-dimensional spaces, but they can be manipulated by using analogies with figures in the plane or in three-dimensional space. In the present chapter we add to the already built arsenal of geometric tools yet another one: the study of convex sets by means of their extreme points. We demonstrate Krein–Milman theorem on existence of extreme points in convex compact sets and give a number of applications, in particular the proof of the Stone–Weierstrass theorem invented by de Branges, Choquet’s proof of Bernstein’s representation for completely monotone functions, and Lindenstrauss’ proof of Lyapunov’s theorem on vector measures.

Vladimir Kadets

Backmatter

Titel: A Course in Functional Analysis and Measure Theory
verfasst von: Prof. Vladimir Kadets
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-92004-7
Print ISBN: 978-3-319-92003-0
DOI: https://doi.org/10.1007/978-3-319-92004-7