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A Guide to Advanced Real Analysis

A Guide to Advanced Real Analysis

Gerald B. Folland
Volume: 37
Copyright Date: 2009
Edition: 1
Pages: 118
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  • Book Info
    A Guide to Advanced Real Analysis
    Book Description:

    A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.

    eISBN: 978-0-88385-915-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-viii)
    Gerald B. Folland
  3. Table of Contents
    (pp. ix-x)
  4. Prologue: Notation, Terminology, and Set Theory
    (pp. 1-4)

    In this prologue we set the stage by briefly discussing some points of notation and terminology and a few facts from set theory that will be used throughout the book.

    We set

    $\mathbb{N}$– the set of positive integers,

    $\mathbb{Z}$= the set of integers,

    $\mathbb{R}$= the set of real numbers,

    $\mathbb{C}$– the set of complex numbers.

    We often enlarge the real number system by adjoining two “elements at infinity,”$\infty $(also called$ + \infty $for emphasis) and$ - \infty $. In the extended system$\mathbb{R} \cup \{ \pm \infty \} = [ - \infty ,\infty ]$, every setEhas a least upper bound orsupremumand a greatest lower bound orinfimum,...

  5. CHAPTER 1 Topology
    (pp. 5-20)

    The subject of this chapter ispoint-set topologyorgeneral topology, the abstract mathematical framework for the study of limits, continuity, and the related geometric properties of sets.

    The early years of the twentieth century witnessed a great increase in the level of abstraction and generality in mathematical thinking. In particular, mathematicians at that time developed theories that provide a very general setting for studying the circle of ideas related to limits and continuity, which previously had been considered in the context of subsets of Euclidean space or functions of one or several real or complex variables.

    The most straightforward...

  6. CHAPTER 2 Measure and Integration: General Theory
    (pp. 21-40)

    The theory of measure of subsets of Euclidean space (length, area, volume, and their analogues in higher dimensions) and the closely related theory of integration of functions on Euclidean space have a very long history. Much of the modern theory, however, does not depend on the particular features of the geometry of Euclidean space. It can be developed in a much more general setting with no additional effort, and in this more general form it yields results that can be applied in many additional situations.

    This abstract theory is the subject of the present chapter; the methods for constructing interesting...

  7. CHAPTER 3 Measure and Integration: Constructions and Special Examples
    (pp. 41-62)

    In this chapter we begin by presenting a general scheme for constructing measures. We then use it to construct Lebesgue measure and related measures on Euclidean space, and we analyze these measures and their associated integrals in some detail. We conclude with a discussion of regular Borel measures and integrals on locally compact Hausdorff spaces.

    The construction of nontrivial examples of measures is not easy. To motivate the ideas, let us consider the elementary notion of area for regions in the plane${\mathbb{R}^2}$that is defined in terms of grids of rectangles. We first define the area of rectangle (the...

  8. CHAPTER 4 Rudiments of Functional Analysis
    (pp. 63-74)

    Functional analysis — the meeting ground of analysis and linear algebra, mostly in an infinite-dimensional setting — is a vast subject, and this brief account does no more than scratch the surface. Our object is simply to introduce some basic concepts that are of wide utility and a few fundamental theorems: just enough to support the material in the last two chapters of this book. For those who want to learn more there are many books available; Reed and Simon [14] and Rudin [18] are among the best.

    Letxbe a vector space over the field$\mathbb{F}$, where$\mathbb{F}$is either...

  9. CHAPTER 5 Function Spaces
    (pp. 75-84)

    In this chapter we study some of the spaces of functions that are of fundamental importance in modern analysis.

    Let$(X,M,\mu )$be a measure space. We recall that${L^1}(\mu )$is the space of allμ-integrable complex-valued functions onX. For$0 < P < \infty $, we define the space${L^p}(\mu )$(also denoted by${L^p}(X)$or simply${L^p}$whenμis understood) to be the set of all measurable complex-valued functionsfonXsuch that${\left| f \right|^p} \in {L^1}(\mu )$. Thus, with the notation

    (5.1)${\left\| f \right\|_p} = {\left[ {\int {{{\left| f \right|}^p}d\mu } } \right]^{1/p}}$,

    ${L^p}(\mu )$is the space of all measurable functions onXsuch that${\left\| f \right\|_p} < \infty $. As in the case of${L^1}(\mu )$, two functions are...

  10. CHAPTER 6 Topics in Analysis on Euclidean Space
    (pp. 85-100)

    In this chapter we present a few basic applications of the abstract ideas and results from the preceding chapters in the concrete setting of the analysis of functions of one or several real variables. There is much more to be said; we are merely scratching the surface of a vast subject that has undergone a vigorous development in the last century. Some references for more extensive treatments include Dym and McKean [2], Strichartz [21], and Stein [20].

    We begin with a few matters of notation. First, we shall denote the integral of a functionfon${\mathbb{R}^n}$with respect to...

  11. Bibliography
    (pp. 101-102)
  12. Index
    (pp. 103-106)
  13. Back Matter
    (pp. 107-107)