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This book is the first of a three volume introduction to analysis. It is distinguished by its modern and clear presentation, concentrating always on the essential concepts. In contrast to most other textbooks, there is no artificial separation between the theories of one variable and that of many variables. Emphasis is placed on the early development of a solid foundation in topology. As well, the basics of complex analysis are covered.This book is directed primarily to the students and instructors of beginning courses in analysis. But, with the many examples, exercises and the supplementary material, it is also suitable for self-study, as preparation for advanced study, and as a basis for other research in mathematics and physics."This textbook provides an outstanding introduction to analysis. It is distinguished by its high level of presentation and its focus on the essential.''Zeitschrift für Analysis und ihre Anwendung 18, No. 4 (G. Berger, review of the first German edition)"One advantage of this presentation is that the power of the abstract concepts are convincingly demonstrated using concrete applications.''W. Grölz, review of the first German edition
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.
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.
"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