Introduction to Modern Analysis: Math 523H, Spring 2017

Class Meeting : TuTh 11:30--12:45, LGRT 177

Instructor : Luc Rey-Bellet

Office :  1423 J LGRT
Phone :  545-6020
E-Mail :   luc <at> math.umass.edu
Homepage:   http://www.math.umass.edu/~lr7q
Office Hours :   Th 9:30-10:30   Fr 10:00--11:30,   or by appointment.

Course Web page: Please bookmark www.math.umass.edu/~lr7q/m523-spring2017/m523home.html The page will be updated regularly. Check it often.

Syllabus: This course is the first part of the Introduction to Analysis sequence (Math 523 and Math 524). At its core analysis deals with mathematical objects such as functions, limits, sequences, series and integrals. The ideas and objects in analysis play an important role in differential equations, probability, numerical analysis, geometry, and in most areas of applied mathematics. But in studying analysis one learns that mathematics is much more than just a set of methods that work. Students will be asked to construct proofs, sometimes long ones, and this will open the door to a much deeper understanding of the nature of mathematics. In Math 523 we deal mostly with "classical analysis" which is a set of ideas and objects formalized in the 19th century. Modern analysis (roughly speaking early 20th century) objects will appear in Math 524. Topics covered in this class are

• Sequences, limits, accumulations points, series . This is a central object in analysis and in studying it we will revisit the concept of real number. One the highlight is the concept of completeness. We also study series (i.e. infinite sums) or real numbers.

• Continuity. We introduce the concepts of a continuous function and of uniform continuity and then combine this with the concept of limits to study limits of function.

• Sequences and series of functions. We combine both previous topics to study power series and introduce the concept of uniform convergence.

• Differentiation. We study differentiation, prove the mean value theorem, L'Hospital theorem and Taylor expansion of a function.

• Integration. We study (Rieman) integrals and the fundamental theorem of calculus.

Textbooks and references: We will follow the textbook only loosely. There are many textbooks to chose from on the topic. The official textbook is suitable for self-learning for "beginners". It is highly recommended you borrow another text and spend some time reading through it.

Official Textbook:

• Elementary Analysis: The Theory of Calculus by Kenneth Ross. 2nd edition Springer-Verlag, 2013. ISBN-13: 000-1461462703 ISBN-10: 1461462703

Other recommended textbooks:

• Fundamental Ideas of Analysis by Michael Reed, Wiley, 1998. ISBN-13: 978-0471159964 ISBN-10: 0471159964. An excellent, but pricey book.

• Introduction to Real Analysis by William Trench. This book can be freely and legally downloaded HERE

• Analysis by its History by by Ernest Hairer and Gerhard Wanner. Springer, 1996. ISBN-13: 978-0387770314 ISBN-10: 0387770313. A wonderful book with a wider scope and a wealth of historical information. The material we covered is essentially Chapter 3 of the book. Highly recommended.

• A radical approach to real analysis by David Bressoud. MAA, 2006 ISBN-10: 0883857472 ISBN-13: 978-0883857472. A great book with historical insights about how analysis evolved. Highly recommended.

Grading and Exams: Your grade will be based on homework's (%30), a mid-term exam (%30), a final exam (%30) and class attendance (%10).

• Midterm Exam : Wednesday 3/1, 7:00--9:00 pm, LGRT 177

• Final exam : Tuesday 5/9/2017, 1:00--3:00 LGRT 177

Homework: Homework will be assigned regularly (essentially every week) and graded.