Real and Functional Analysis / Edition 3

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This book is meant as a text for a first-year graduate course in analysis. In a sense, it covers the same topics as elementary calculus but treats them in a manner suitable for people who will be using it in further mathematical investigations. The organization avoids long chains of logical interdependence, so that chapters are mostly independent. This allows a course to omit material from some chapters without compromising the exposition of material from later chapters.

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Editorial Reviews

The second edition was published as Real Analysis, Addison- Wesley, 1983. The third edition has been reorganized. After a brief introduction to point set topology, some basic theorems on continuous functions, and the introduction of Banach and Hilbert spaces, integration theory is covered so it can fit in a one-semester course. Functional analysis follows for a second semester. Examples and exercises have been added, as well as material on integration pertaining to the real line. Functional analysis now includes the Gelfand transform, and basic spectral theorems for compact, bounded hermitian, and unbounded self-adjoint operators. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780387940014
  • Publisher: Springer New York
  • Publication date: 5/3/1993
  • Series: Graduate Texts in Mathematics Series, #142
  • Edition description: 3rd ed. 1993
  • Edition number: 3
  • Pages: 580
  • Product dimensions: 1.38 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Ch. ISets3
1Some Basic Terminology3
2Denumerable Sets7
3Zorn's Lemma10
Ch. IITopological Spaces17
1Open and Closed Sets17
2Connected Sets27
3Compact Spaces31
4Separation by Continuous Functions40
Ch. IIIContinuous Functions on Compact Sets51
1The Stone-Weierstrass Theorem51
2Ideals of Continuous Functions55
3Ascoli's Theorem57
Ch. IVBanach Spaces65
1Definitions, the Dual Space, and the Hahn-Banach Theorem65
2Banach Algebras72
3The Linear Extension Theorem75
4Completion of a Normed Vector Space76
5Spaces with Operators81
Ch. VHilbert Space95
1Hermitian Forms95
2Functionals and Operators104
Ch. VIThe General Integral111
1Measured Spaces, Measurable Maps, and Positive Measures112
2The Integral of Step Maps126
3The L[superscript 1]-Completion128
4Properties of the Integral: First Part134
5Properties of the Integral: Second Part137
7Extension of Positive Measures from Algebras to [sigma]-Algebras153
8Product Measures and Integration on a Product Space158
9The Lebesgue Integral in R[superscript p]166
Ch. VIIDuality and Representation Theorems181
1The Hilbert Space L[superscript 2](mu)181
2Duality Between [actual symbol not reproducible]185
3Complex and Vectorial Measures195
4Complex or Vectorial Measures and Duality204
5The L[superscript p] Spaces, [actual symbol not reproducible]209
6The Law of Large Numbers213
Ch. VIIISome Applications of Integration223
2Continuity and Differentiation Under the Integral Sign225
3Dirac Sequences227
4The Schwartz Space and Fourier Transform236
5The Fourier Inversion Formula241
6The Poisson Summation Formula243
7An Example of Fourier Transform Not in the Schwartz Space244
Ch. IXIntegration and Measures on Locally Compact Spaces251
1Positive and Bounded Functionals on C[subscript c](X)252
2Positive Functionals as Integrals255
3Regular Positive Measures265
4Bounded Functionals as Integrals267
5Localization of a Measure and of the Integral269
6Product Measures on Locally Compact Spaces272
Ch. XRiemann-Stieltjes Integral and Measure278
1Functions of Bounded Variation and the Stieltjes Integral278
2Applications to Fourier Analysis287
Ch. XIDistributions295
1Definition and Examples295
2Support and Localization299
3Derivation of Distributions303
4Distributions with Discrete Support304
Ch. XIIIntegration on Locally Compact Groups308
1Topological Groups308
2The Haar Integral, Uniqueness313
3Existence of the Haar Integral319
4Measures on Factor Groups and Homogeneous Spaces322
Ch. XIIIDifferential Calculus331
1Integration in One Variable331
2The Derivative as a Linear Map333
3Properties of the Derivative335
4Mean Value Theorem340
5The Second Derivative343
6Higher Derivatives and Taylor's Formula346
7Partial Derivatives351
8Differentiating Under the Integral Sign355
9Differentiation of Sequences356
Ch. XIVInverse Mappings and Differential Equations360
1The Inverse Mapping Theorem360
2The Implicit Mapping Theorem364
3Existence Theorem for Differential Equations365
4Local Dependence on Initial Conditions371
5Global Smoothness of the Flow376
Ch. XVThe Open Mapping Theorem, Factor Spaces, and Duality387
1The Open Mapping Theorem387
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