Math 508: Advanced Analysis I |
Fall 2014 |
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Faculty: Jerry L. KazdanTelephone: (215) 898-5109

email:kazdan AT math.upenn.eduOffice Hours: Wed. 10:30-11:30 (and by appointment) in DRL 4E15TA: Spencer Tofts, DRL 3E2Telephone: (215) 898-8175

email:spencertofts AT gmail.comOffice Hours: Mon. and Wed. 4:30-5:30 (and by appointment) in DRL 3E2

Text:Walter Rudin,Principles of Mathematical Analysis, McGraw-Hill, 1976 (3rd Edition, ISBN:9780070542358.

An electronic version seems to be available online for free.The course will mainly follow the order of topics in the Rudin text.

Solutions to problems in Rudin: In a simple search, I located two sources of solutions to the problems in the Rudin text

Rudin Solutions (Cooke) and Rudin Solutions (Bergman)

Also: (Bergman) Remarks and Errata on Rudin, Rudin: Answers to questions asked by students

In Fall 2006, 2008 and 2010 I taught a similar course. See the web pages:

Math 508: Fall 2006, Math 508: Fall 2008, Math 508: Fall 2010,Prerequisites & Review Material

Exams:There will be two in-class exams, from 9:00-10:20 on

- Exam 1, Tues. Oct. 14: PDF condensed Solutions
- Exam 2, Tues. Dec. 9: PDF condensed Solutions
- No Final Exam
- Exam Scores
You may always use one 3"x5" card with notes on both sides

Some References:books, articles, web pagesNote that a Canvas site has been arranged for the course. You may find it useful for communication.

Step one is to sign-up.

Notes:

LaTeX: If you will be writing many documents that contain equations, it is wise to learn (and use) LaTeX. It is available on Windows, Macs, and Linux -- and isfree. See TeX Stuff. For some students, this might be the most useful item you learn in this course.

Aug 28

Field Axioms

Reals: Archimedean property [LaTeX source]

Notes for Sept. 4, 2014 [LaTeX source]

Some Basic Definitions [LaTeX source] [large]

Some Basic Examples [ LaTeX source] [large]

Relative Topology

Compactness [LaTeX source] [large]

Logic notation: Convergence and Continuity [LaTeX source]

Mobius Strip (Maple 15)

Completeness of l_{_1}

Cobweb Model

Dirichlet's Principle [LaTeX Source] [large]

Compactness and Uniform Continuity [ LaTeX Source]

Taylor's Theorem: Integral Remainder [LaTeX Source] [large]

Differentiation of a limit

Contracting Mappings (K-F) Existence: u'=Au + f ODE Systems Existence: Marsden-Hoffman

Notes on Convolution [large]

Some Puzzles

Remark: The Homework Assignments:difficultyof an exercise is not at all the same as theamount of work it involves. A long series of straightforward manipulations can have a low level of difficulty, but involve a lot of work. In this course the majority of problems will have short solutions -- although it may take real time to think them through.

- Set 0 Due: Thurs., Sept. 4 (Rust Remover. These problems mainly involve only
High Schoolmathematics. They will not be collected.)- Set 1 Due: Tues., Sept. 9. Late papers: accepted until 1:00 PM Wednesday [LaTeX source]
- Set 2 Due: hurss., Sept. 11. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 3 Due: Thurs., Sept. 18. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 4 Due: Thurs., Sept. 25. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 5 Due: Thurs., Oct. 2. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 6 Due: Thurs., Oct. 23. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 7 Due: Thurs., Oct. 30. Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 8 Due: Thurs., Nov. 6 Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 9 Due: Thurs., Nov. 13 Late papers: accepted until 1:00 PM Friday [LaTeX source]
- Set 10 Due: Thurs., Nov. 20 Late papers: accepted until 1:00 PM Friday [LaTeX source]

Solution of Maximal Ideals Bonus Problem- Set 11 Due: Tues., Dec. 2 (lLate papers: accepted until 1:00 PM Wednesday) [LaTeX source]

Solution of hw 11 Bonus #1

Old Exams:

Fall 2010 Exam 1 Fall 2010 Exam 2 [Exam 2 solutions]

Fall 2008 Exam 1 Fall 2008 Exam 2 [solutions]

Fall 2006 Exam 1 [solutions] Fall 2006 Exam 2 [solutions].