Prerequisites:  Math 401 and 402 (Advanced Calculus I & II) or equivalent.  Or consent of the instructor.

Course Description:  A graduate introduction to real analysis.  We will cover the following topics corresponding to chapters of the text:

Set Theory (1)  quick review Real Numbers (2) review Lebesgue Measure (3) core

Lebesgue Integral (4) core Differentiation (5) lighter Lp Spaces (6) core

Metric Spaces (7) as needed Banach&Hilbert Spaces (10) General Measure Theory (11)

Class time will be spent half on my lecture and half on problem sessions.

There will be weekly homework assignments, usually with 10 or fewer problems.  Firmly due at the date given.

There will be two midterm quizzes, done in class.  These will include questions like "What is the definition of ..." and also problems whose solution requires a routine use of the definitions.  They will be easier than the homework questions.

There will be a take-home Final exam.  It will be very firmly due on the given date.

Text:  H. L. Royden, Real Analysis, Macmillan/Prentice Hall 3rd ed., 1988.

But the following are recommended and in the Rasmuson Library:

• Chae, Lebesgue Integration.  Sticks to the one dimensional case, and may be a good second reading for this course.
• Kolmogorov & Fomin, Introductory Real Analysis (english translation available  from Dover).  Organizes things differently.  Only gives real examples at the end--but a great book.
• Lang, Real and Functional Analysis.  General.  Topological.  Makes clear that the theory of vector spaces of functions is central.
• Taylor, An Introduction to Measure and Probability.  A crucial realization for grad students: Probability implies Lebesgue integration. (Or: the best single argument for the Lebesgue theory is the unification of probability and analysis.)
• Rudin, Real & Complex Analysis.  One of the standards.  Fast moving.  Always goes right to the most general case.