# Real and Functional Analysis

• Serge Lang
Book

Part of the Graduate Texts in Mathematics book series (GTM, volume 142)

1. Front Matter
Pages i-xiv
2. ### General Topology

1. Front Matter
Pages 1-1
2. Serge Lang
Pages 3-16
3. Serge Lang
Pages 17-50
4. Serge Lang
Pages 51-62
3. ### Banach and Hilbert Spaces

1. Front Matter
Pages 63-63
2. Serge Lang
Pages 65-94
3. Serge Lang
Pages 95-108
4. ### Integration

1. Front Matter
Pages 109-110
2. Serge Lang
Pages 111-180
3. Serge Lang
Pages 181-222
4. Serge Lang
Pages 223-250
5. Serge Lang
Pages 251-277
6. Serge Lang
Pages 278-294
7. Serge Lang
Pages 295-307
8. Serge Lang
Pages 308-328
5. ### Calculus

1. Front Matter
Pages 329-329
2. Serge Lang
Pages 331-359
3. Serge Lang
Pages 360-384
6. ### Functional Analysis

1. Front Matter
Pages 385-386

### Introduction

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.

### Keywords

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

#### Authors and affiliations

• Serge Lang
• 1
1. 1.Department of MathematicsYale UniversityNew HavenUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0897-6
• Copyright Information Springer-Verlag New York, Inc. 1993
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-6938-0
• Online ISBN 978-1-4612-0897-6
• Series Print ISSN 0072-5285