Undergraduate Analysis

1. Front Matter

   Pages i-xv

   PDF

2. Review of Calculus

   1. Front Matter

      Pages 1-1

      PDF

   2. Sets and Mappings

      • Serge Lang

      Pages 3-16

   3. Real Numbers

      • Serge Lang

      Pages 17-33

   4. Limits and Continuous Functions

      • Serge Lang

      Pages 34-65

   5. Differentiation

      • Serge Lang

      Pages 66-77

   6. Elementary Functions

      • Serge Lang

      Pages 78-100

   7. The Elementary Real Integral

      • Serge Lang

      Pages 101-125

3. Convergence

   1. Front Matter

      Pages 127-128

      PDF

   2. Normed Vector Spaces

      • Serge Lang

      Pages 129-159

   3. Limits

      • Serge Lang

      Pages 160-192

   4. Compactness

      • Serge Lang

      Pages 193-205

   5. Series

      • Serge Lang

      Pages 206-245

   6. The Integral in One Variable

      • Serge Lang

      Pages 246-279

4. Applications of the Integral

   1. Front Matter

      Pages 281-282

      PDF

   2. Approximation with Convolutions

      • Serge Lang

      Pages 283-290

   3. Fourier Series

      • Serge Lang

      Pages 291-325

   4. Improper Integrals

      • Serge Lang

      Pages 326-352

   5. The Fourier Integral

      • Serge Lang

      Pages 353-367

5. Calculus in Vector Spaces

   1. Front Matter

      Pages 369-370

      PDF

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

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

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

  Serge Lang

Policies and ethics