Applied Analysis Preliminary Examination, Format/Grading, Content Bins and Expectations of Students
The exam will be 8 questions split into two parts with questions coming from the syllabus that is aligned with the content bins discussed below.
The first part of the exam will have 4 questions, one from each of the content bins. Problems in this part of the exam are generally more direct and self-contained within each of the content bins. Students should answer all questions in this part of the exam.
The second part of the exam will have 4 questions covering more advanced material or more integrated material within or across the content bins. Students are required to choose 2 of the 4 questions to answer in this part of the exam.
The grading of the exam and determination of pass/fail is based upon student responses for both parts of the exam. Solutions (proofs, examples/counterexamples, etc.) are graded on both correctness of the mathematical argument and correct writing. Extraneous material provided by the student that is not relevant to a solution negatively impacts the assessment of the solution. Progress towards the problem as stated is required for partial credit; solutions of a similar related or a partial problem are considered not relevant. Incorrect or imprecise notation, or lack of proper mathematical proof writing negatively impacts the assessment of the solution.
While textbooks and courses may differ somewhat between years when this course is offered, the general content is as follows, based on Rudin’s Principles of Mathematical Analysis, McGraw-Hill, 3rd edition, 1976, for reference:
Real and complex numbers, infimum, supremum: Rudin 1.1-1.20, 1.23-1.38, Ch. 1 exercises 1-5, 8-18.
Metric spaces and real line topology: Rudin 2.1-2.42, 2.45-2.47, Ch. 2 exercises 1-16, 19-27
Continuous functions in metric spaces and on reals: Rudin 4.1-4.19, 4.25-4.33, Ch. 4 exercises 1-
6, 8-18, 20-25.
Complete and compact metric space, numerical sequences and series, upper and lower limits,
power series: Rudin 3.1-3.49, 8.1-8.3, Ch. 3 exercises 1-11, 20, 21, 23-25, Ch. 8 exercises 1-3 Continuity and derivatives of functions of real variable, mean value theorem: Rudin 5.1-5.11, Ch. 5 exercises 1-6
Riemann Integration: Rudin 6.1-6.10, 6.12, Ch. 6 exercises 1-5.
Sequences and series of functions, uniform convergence and switching limits, Arzèla-Ascoli theorem, Stone-Weierstrass theorem: Rudin 7.1-7.26, Ch. 7 exercises 1-9, 11, 15-20, 24
General expectations of students
Students should be able to write clear, correct, and concise proofs based on definitions and to use standard techniques where applicable. Students should be able to prove standard results in each of the content bins. Students should also be able to provide simple, straightforward counterexamples to false statements (e.g., by the removal of a necessary condition in a standard result), and students are expected to justify their counterexamples.
Students should be able to solve problems involving both abstract metric spaces and canonical examples of metric spaces including Rn (with Euclidean metrics), C ([a,b]) (with sup-norm or integral metrics), and the discrete metric. Students should also be able to solve problems involving mappings between metric spaces and use calculus concepts as necessary and where applicable.