Real and Functional Analysis

1. Front Matter

   Pages i-xiv

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2. General Topology

   1. Front Matter

      Pages 1-1

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   2. Sets

      • Serge Lang

      Pages 3-16

   3. Topological Spaces

      • Serge Lang

      Pages 17-50

   4. Continuous Functions on Compact Sets

      • Serge Lang

      Pages 51-62

3. Banach and Hilbert Spaces

   1. Front Matter

      Pages 63-63

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   2. Banach Spaces

      • Serge Lang

      Pages 65-94

   3. Hilbert Space

      • Serge Lang

      Pages 95-108

4. Integration

   1. Front Matter

      Pages 109-110

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   2. The General Integral

      • Serge Lang

      Pages 111-180

   3. Duality and Representation Theorems

      • Serge Lang

      Pages 181-222

   4. Some Applications of Integration

      • Serge Lang

      Pages 223-250

   5. Integration and Measures on Locally Compact Spaces

      • Serge Lang

      Pages 251-277

   6. Riemann-Stieltjes Integral and Measure

      • Serge Lang

      Pages 278-294

   7. Distributions

      • Serge Lang

      Pages 295-307

   8. Integration on Locally Compact Groups

      • Serge Lang

      Pages 308-328

5. Calculus

   1. Front Matter

      Pages 329-329

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   2. Differential Calculus

      • Serge Lang

      Pages 331-359

   3. Inverse Mappings and Differential Equations

      • Serge Lang

      Pages 360-384

6. Functional Analysis

   1. Front Matter

      Pages 385-386

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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.

• Banach Space
• Distribution
• Hilbert space
• calculus
• differential equation
• functional analysis
• measure
• spectral theorem

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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