Applied Analysis Preliminary Exam Syllabus

***DRAFT***

The analysis preliminary exam is designed to test students' knowledge of basic concepts associated with the study of functions of real variables and their ability to formulate proofs of assertions about these basic concepts. The primary references are

 Both books cover all required topics. Students can use either one, with few exceptions noted.

  • Real numbers, infimum, supremum: Rudin 1.1-1.20, 1.23-1.38,  or Schröder 1
  • Real Line and Metric Space Topology: Rudin 2.1-2.42, 2.45-2.47, or Schröder  16.1-16.8, except Thm. 6.19
  • Numerical Sequences and Series: Rudin 3.1-3.55, or Schröder 2, 6, 10.1, 10.2
  • Continuous Functions: Rudin 4.1-4.19, 4.25-4.33, or Schröder 15.1-15.7, 16
  • Differentiation: Rudin 5.1-5.19, or Schröder  4, 17.1-17.5
  • Riemann Integration: Rudin 6.1-6.9, 6.12-6.18, 6.20-6.27, Schröder 5, exercises 5-12, 5-19, 5-27 (Riemann-Stieltjes integral) 
  • Sequences and Series of Functions: Rudin 7.1-7.26, or Schröder 11 and Rudin 7.23-7.26
  • Power Series: Rudin 8.1-8.5, or Schröder 10.3
  • Functions of Several Variables: Rudin 9.1-9.29, 9.39-9.42, or Schröder  17.1-17.5, 17.8

Further required material is in the following recommended exercises. You may need to consult other books to solve some of the exercises. All exercises from Schröder in covered sections are suitable for this level. Many exercises from Rudin are much harder. Selected suitable exercises from Rudin:

  • Rudin Ch. 1 exercises 1-5, 8-19
  • Rudin Ch. 2 exercises 1-16, 19-27, 29
  • Rudin Ch. 3 exercises 1-18, 20, 21, 23-25
  • Rudin Ch. 4 exercises 1-6, 8-18, 20-25
  • Rudin Ch. 5 exercises 1-7, 9-14, 19, 20, 22-24
  • Rudin Ch. 6 exercises 1-5
  • Rudin Ch. 7 exercises 1-13,15-20,24
  • Rudin Ch. 8 exercises 1-5,7-9
  • Rudin Ch. 9 exercises 1-8,9 (for convex set), 11-16,20,27,30,31

We also recommend the following supplemental texts, which contain the same topics covered from different vantages and some at a greater depth.

  • Patrick M. Fitzpatrick, Advanced Calculus: A Course in Mathematical Analysis, PWS, Boston, MA, 1996.
  • Robert C. Buck and Ellen F. Buck, Advanced Calculus, McGraw-Hill, New York, NY, 1978.
  • Maxwell Rosenlicht, Introduction to Analysis, Dover, 1986.
  • Georgi E. Shilov, Elementary Real and Complex Analysis, (Ch. 1-9), Dover, revised edition, 1996.
  • Herbert S. Gaskill and P. P. Narayanaswami, Elements of Real Analysis, Prentice Hall 1997.

Rudin's book is the best source, but many problems from it are too long to be useful practice problems for the prelim exam. Shorter problems from other references are more appropriate. The student exam paper must be written in a way that the evaluators can understand clearly the student's line of argument so that the correctness of the argument can be decided. Please do not write incorrect statements, hoping for partial credit. The evaluators will not fill in missing arguments, make educated guesses about what the student had in mind, or patch a solution from correct statements mixed with incorrect ones. The expected style of the solutions is essentially the same as as seen in graduate texbooks, in monographs, or in papers. In particular:

  • The text and formulas, when read aloud, should form complete correct English sentences.
  • Mathematical statements should be complete. There should be no fragments.
  • It should be indicated how each statement is justified. For example, if it follows from the preceding statement words such as "so", "hence", "it follows that" should be used. If it follows from something else, it should be explicity stated from what it follows, such as a well-known theorem or an earlier statement. If a statement is assumed only temporarily, e.g., to discuss one of several cases or to derive a contradiction, it should be indicated.

To prepare for the exam, students are strongly advised to solve as many problems from the recommended texts as possible, in particular the recommended exercises above, try to find counterexamples to theorems when some assumptions are dropped, and practice writing their solutions in the required style. The classes Math 5070 - Applied Analysis , Math 4310 - Introduction to Real Analysis I , and Math 4320 - Introduction to Real Analysis II can help with preparation for this exam. However, students should note that these classes are not for PhD students only, so the grading of students' work in these classes cannot reflect the standards of mathematical accuracy and expression expected on this exam. Further, these courses do not cover all of the required material.