Penn Math Math 508: Advanced Analysis I Fall 2014

Faculty: Jerry L. Kazdan
Telephone: (215) 898-5109
email: kazdan AT
Office Hours: Wed. 10:30-11:30   (and by appointment) in DRL 4E15
TA: Spencer Tofts, DRL 3E2
Telephone: (215) 898-8175
email: spencertofts AT  
Office 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

You may always use one 3"x5" card with notes on both sides

Course and Homework Grading

Some References: books, articles, web pages

Note that a Canvas site has been arranged for the course. You may find it useful for communication. Step one is to sign-up.

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 is free. 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

Homework Assignments:
Remark: The difficulty of an exercise is not at all the same as the amount 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.
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].