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1993 | Buch | 3. Auflage

Sets Kapitel lesen Erstes Kapitel lesen

Real and Functional Analysis

verfasst von: Serge Lang

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

This book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding, for instance, my Undergraduate Anal­ ysis. I assume that the reader is acquainted with notions of uniform con­ vergence and the like. In this third edition, I have reorganized the book by covering inte­ gration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises, as well as some material about integration on the real line (e.g. on Dirac sequence approximation and on Fourier analysis), and some material on functional analysis (e.g. the theory of the Gelfand transform in Chapter XVI). These upgrade previous exercises to sections in the text. In a sense, the subject matter covers the same topics as elementary calculus, viz. linear algebra, differentiation and integration. This time, however, these subjects are treated in a manner suitable for the training of professionals, i.e. people who will use the tools in further investiga­ tions, be it in mathematics, or physics, or what have you. In the first part, we begin with point set topology, essential for all analysis, and we cover the most important results.

Inhaltsverzeichnis

Frontmatter

General Topology

Frontmatter
Chapter I. Sets
Abstract
We assume that the reader understands the meaning of the word “set”, and in this chapter, summarize briefly the basic properties of sets and operations between sets. We denote the empty set by Ø. A subset S′ of S is said to be proper if S′ ≠ S. We write S′ ⊂S or S ⊃ S′ to denote the fact that S′ is a subset of S.
Serge Lang
Chapter II. Topological Spaces
Abstract
This chapter develops the standard properties of topological spaces. Most of these properties do not go beyond the level of a convenient language. In the text proper, we have given precisely those results which are used very frequently in all analysis. In the exercises, we give additional results, of which some just give routine practice and others give more special results. To incorporate all this material in the text proper would be extremely oppressive and would obscure the principal lines of thought inherent in the basic aspects of the subject. The reader can always be referred to Bourbaki [Bo] or Kelley [Ke] for encyclopaedic treatments.
Serge Lang
Chapter III. Continuous Functions on Compact Sets
Abstract
Let E be a normed vector space (over the real or the complex numbers). We can define the notion of Cauchy sequence in E as we did for real sequences, and also the notion of convergent sequence (having a limit). If every Cauchy sequence converges, then E is said to be complete, and is also called a Banach space. A closed subspace of a Banach space is complete, hence it is also a Banach space.
Serge Lang

Banach and Hilbert Spaces

Frontmatter
Chapter IV. Banach Spaces
Abstract
Let E be a Banach space, i.e. a complete normed vector space. One can deal with series ∑ x n in Banach spaces just as with series of numbers, or of functions, and the most frequent test for convergence (in fact absolute convergence) is the standard one: Let {a n } be a sequence of numbers ≧ 0 such that ∑ a n converges. If |x n | a n for all n, then ∑ x n converges.
Serge Lang
Chapter V. Hilbert Space
Abstract
Essentially all of this chapter goes through over the real or the complex numbers with no change. Since the theory over the complex does introduce the extra conjugation, we use the complex language, and point out explicitly in one or two instances those results which are valid only over the complex.
Serge Lang

Integration

Frontmatter
Chapter VI. The General Integral
Serge Lang
Chapter VII. Duality and Representation Theorems
Abstract
Consider first complex valued functions. We let ℒ2(μ) be the set of all functions f on X that are limits almost everywhere of a sequence of step functions (i.e. μ-measurable), and such that |f|2 lies in ℒ1. Thus
$$ {\left| f \right|^{2}} = \overline {ff} . $$
Serge Lang
Chapter VIII. Some Applications of Integration
Abstract
After the abstract theory on arbitrary measured spaces, it is a relief to get into some classical situations on R n where we see the integral at work. None of this chapter will be used later, except for the approximation by Dirac families in the uniqueness proof for the spectral measure of Chapter 20.
Serge Lang
Chapter IX. Integration and Measures on Locally Compact Spaces
Abstract
On a locally compact space, it is as natural to deal with continuous functions having compact support as it is natural to deal with step functions. Thus we must establish the relations which exist between functionals on the former or the latter. As we shall see, they essentially amount to the same thing.
Serge Lang
Chapter X. Riemann-Stieltjes Integral and Measure
Abstract
This chapter gives an example of a measure which arises from a functional, defined essentially by generalizations of Riemann sums. We get here into special aspects of the real line, as distinguished from the general theory of integration on general spaces.
Serge Lang
Chapter XI. Distributions
Abstract
In Chapter IX, we saw how certain functional on C c (X) gave rise to a measure. Here we consider the case when X = R n and the functionals satisfy additional continuity conditions with respect to differentiation.
Serge Lang
Chapter XII. Integration on Locally Compact Groups
Abstract
This chapter is independent of the others, but is interesting for its own sake. It gives examples of integration in a different setting from euclidean space, for instance integration on a group of matrices. For an application of integration and some functional analysis to compact groups, see Exercise 11.
Serge Lang

Calculus

Frontmatter
Chapter XIII. Differential Calculus
Abstract
Let [a, b] be a closed interval, and E a Banach space. By a step map f: [a, b] → E we mean a map for which there exists a partition P:a = a 0 ≦ a 1 ≦... ≦ a n = b and elements v 1, ...,v n ∈ E such that if a i-1 < t < a i , then f( t) = v i .
Serge Lang
Chapter XIV. Inverse Mappings and Differential Equations
Abstract
Both the inverse mapping theorem and the existence theorem for differential equations will be based on a basic and simple lemma in complete metric spaces.
Serge Lang

Functional Analysis

Frontmatter
Chapter XV. The Open Mapping Theorem, Factor Spaces, and Duality
Abstract
We begin with a general theorem on metric spaces.
Serge Lang
Chapter XVI. The Spectrum
Abstract
In this chapter we give basic facts about the spectrum of an element in a Banach algebra. Under certain circumstances, we represent such an algebra as the algebra of continuous functions on its spectrum, which is defined as the space of its maximal ideals (or the space of characters), to be given the weak topology. In the next chapters, we shall deal with spectral theorems corresponding to more specific examples of Banach algebras. The proofs in the later chapters are independent of those in the present chapter, except for Theorem 1.2 and its corollaries. Thus the rest of this chapter may be bypassed. On the other hand, the general representation of a Banach algebra as an algebra of continuous functions on the spectrum gives a nice application of the Stone-Weierstrass theorem, and is useful in other contexts besides the spectral theorems which form the remainder of the book, so I have included the basic results to provide a suitable background for applications not included in this book.
Serge Lang
Chapter XVII. Compact and Fredholm Operators
Abstract
The operators in infinite dimensional spaces closest to operators in finite dimensional spaces are the compact operators, which will now be studied systematically. A large number of examples of compact operators are given in the exercises.
Serge Lang
Chapter XVIII. Spectral Theorem for Bounded Hermitian Operators
Abstract
This chapter may be viewed as a direct continuation of the linear algebra in the context of Hilbert space first discussed in Chapter V.
Serge Lang
Chapter XIX. Further Spectral Theorems
Abstract
In this chapter, we use the spectral theorem of Chapter XVIII to give a finer theory, making sense of the expression f(A) when f is not continuous. Ultimately, one wants to use very general functions f in the context of measure theory, namely bounded measurable functions, as a corollary of what was done in Chapter XVIII. For our purposes here, we deal with an intermediate category of functions, essentially characteristic functions of intervals. These give rise to projection operators, whose formalism is important for its own sake. We also want to deal with unbounded operators as an application.
Serge Lang
Chapter XX. Spectral Measures
Abstract
In the spectral theorems of Chapters XVIII and XIX, we defined functions of an operator f(A) with continuous functions first, and then essentially characteristic functions of an interval by a limiting process. If v is a given vector, then the association
$$ f \mapsto \left\langle {f\left( A \right)v,v} \right\rangle $$
defines a functional on C c (R).
Serge Lang

Global Analysis

Frontmatter
Chapter XXI. Local Integration of Differential Forms
Abstract
We recall that a set has measure 0 in R n if and only if, given e, there exists a covering of the set by a sequence of rectangles {R j } such that ∑ μ(R j ) < ε. We denote by R j the closed rectangles, and we may always assume that the interiors R j 0 = Int(R j ) cover the set, at the cost of increasing the lengths of the sides of our rectangles very slightly (an ε2 n argument). We shall prove here some criteria for a set to have measure 0. We leave it to the reader to verify that instead of rectangles, we could have used cubes in our characterization of a set of a measure 0 (a cube being a rectangle all of whose sides have the same length).
Serge Lang
Chapter XXII. Manifolds
Serge Lang
Chapter XXIII. Integration and Measures on Manifolds
Abstract
Throughout this chapter, unless otherwise specified, we use the word manifold to denote manifolds possibly having boundaries. From §3 to the end, we let X be a manifold of class Cp with p ≧ 1, which is Hausdorff and has a countable base. These last two assumptions are to ensure that X admits Cp partitions of unity, subordinated to any given open covering.
Serge Lang
Backmatter
Metadaten
Titel
Real and Functional Analysis
verfasst von
Serge Lang
Copyright-Jahr
1993
Verlag
Springer New York
Electronic ISBN
978-1-4612-0897-6
Print ISBN
978-1-4612-6938-0
DOI
https://doi.org/10.1007/978-1-4612-0897-6