Analysis I

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. Main subject categories: • Mathematical analysis • Real functions • Functions of one variable • Mathematical logic and foundations ‒ Introductory exposition • Sequences, series, summability ‒ Introductory exposition • General topology ‒ Introductory expositionThis 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)

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

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) To make complicated mathematical relationships clear it is convenient to use the notation of symbolic logic.