Analysis I

1. Front Matter

   Pages i-xv

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2. Foundations

   Pages 1-128

3. Convergence

   Pages 129-216

4. Continuous Functions

   Pages 217-298

5. Differentiation in One Variable

   Pages 299-359

6. Sequences of Functions

   Pages 361-403

7. Back Matter

   Pages 405-430

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Logical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems — these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned ‘in a vacuum’. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems.

• Analysis
• calculus
• complex analysis
• convergence
• differential equation

"It is a wonderful book that distinguishes itself by the clarity of presentation, by the fact that it is self-contained and has many exercises at various degrees of difficulty (among many other great qualities.)...[It] is a very elegant and complete treatment of topics usually not found in analysis books...The exercises are plenty and very well selected. In summary, we have here a very-very good, self-contained, well written book, including many applications and exercises, that can be used as a text for courses, as self-study, and/or as a basis for research in mathematics or even other disciplines. This book is a wonderful read." ---MAA Reviews

• Institut für Mathematik, Universität Zürich, Zürich

  Herbert Amann

• Institut für Angewandte Mathematik, Universität Hannover, Hannover

  Joachim Escher

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