© 1997

Undergraduate Analysis


Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Review of Calculus

    1. Front Matter
      Pages 1-1
    2. Serge Lang
      Pages 3-16
    3. Serge Lang
      Pages 17-33
    4. Serge Lang
      Pages 34-65
    5. Serge Lang
      Pages 66-77
    6. Serge Lang
      Pages 78-100
    7. Serge Lang
      Pages 101-125
  3. Convergence

    1. Front Matter
      Pages 127-128
    2. Serge Lang
      Pages 129-159
    3. Serge Lang
      Pages 160-192
    4. Serge Lang
      Pages 193-205
    5. Serge Lang
      Pages 206-245
    6. Serge Lang
      Pages 246-279
  4. Applications of the Integral

    1. Front Matter
      Pages 281-282
    2. Serge Lang
      Pages 283-290
    3. Serge Lang
      Pages 291-325
    4. Serge Lang
      Pages 326-352
    5. Serge Lang
      Pages 353-367
  5. Calculus in Vector Spaces

    1. Front Matter
      Pages 369-370

About this book


This is a logically self-contained introduction to analysis, suitable for students who have had two years of calculus. The book centers around those properties that have to do with uniform convergence and uniform limits in the context of differentiation and integration. Topics discussed include the classical test for convergence of series, Fourier series, polynomial approximation, the Poisson kernel, the construction of harmonic functions on the disc, ordinary differential equation, curve integrals, derivatives in vector spaces, multiple integrals, and others. In this second edition, the author has added a new chapter on locally integrable vector fields, has rewritten many sections and expanded others. There are new sections on heat kernels in the context of Dirac families and on the completion of normed vector spaces. A proof of the fundamental lemma of Lebesgue integration is included, in addition to many interesting exercises.


Analysis Derivative Differentialrechnung Fourier series Integralrechnung calculus compactness differential equation

Authors and affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • Book Title Undergraduate Analysis
  • Authors Serge Lang
  • Series Title Undergraduate Texts in Mathematics
  • DOI
  • Copyright Information Springer-Verlag New York 1997
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-94841-6
  • Softcover ISBN 978-1-4419-2853-5
  • eBook ISBN 978-1-4757-2698-5
  • Series ISSN 0172-6056
  • Edition Number 2
  • Number of Pages XV, 642
  • Number of Illustrations 23 b/w illustrations, 0 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking
IT & Software


Second Edition

S. Lang

Undergraduate Analysis

"[A] fine book . . . logically self-contained . . . This material can be gone over quickly by the really well-prepared reader, for it is one of the book’s pedagogical strengths that the pattern of development later recapitulates this material as it deepens and generalizes it."—AMERICAN MATHEMATICAL SOCIETY