Math 505

Principles of Analysis

Fall Semester 2007

• Class Meets: 11:00-11:50 am, MWF, in Wilson 1-148
• Instructor:
  Curt Vogel
  2-210 Wilson Hall
  ph. 994-5332
  e-mail: vogel (at) math.montana.edu
  Office Hours: 10:00-10:50 MWF
• Textbook
  Principles of Analysis, 3rd Edition, by W. Rudin
  published by McGraw-Hill
• Course Outline:
  Basic Topology
  Numerical Sequences and Series
  Continuity
  Differentiation
  Riemann-Stieltjes Integration
  Sequences and Series of Functions
  Functions of Several Variables
• Homework Assignment 1: Due Friday, Aug. 31
  P. 43 of Rudin, Exercises 1, 4 (explain why or why not), 5, 17 (countability part)
• Homework Assignment 2: Due Friday, Sept. 7
  P. 43 of Rudin, Exercises 6, 7, 9, 12, 14
• Homework Assignment 3: Due Friday, Sept. 13
  P. 44 of Rudin, Exercises 12, 14, 15, 17 (all but countability part), 20, 22
• Homework Assignment 4: Due Friday, Sept. 21
  P. 78 of Rudin, Exercises 1, 2, 3. Hint: Show that f(s)=sqrt(2+sqrt(s)) is monotonic provided sqrt(2) < s < 2.
  Prove that if {p_n} converges to p, then any subsequence of {p_n} also converges to p.
• Homework Assignment 5: Due Friday, Sept. 28
  P. 78 of Rudin, Exercises 7, 8, 9, 10, 14abc.
  State and rigorously prove a self contained version of the integral comparison test. Use the Cauchy criterion in your proof.
  Use your integral comparison test to prove Theorem 3.29.
  Let {p_n} be a bounded, real-valued sequence. Prove that {p_n} converges if and only if lim sup p_n = lim inf p_n.
• Homework Assignment 6: Due Friday, Oct. 5
  P. 98 of Rudin, Exercises 1-4, 7, 11, 14, 20, 21.
• Homework Assignment 7: Due Friday, Oct. 12
  P. 114 of Rudin, Exercises 1-3, 5,7.
• Midterm Exam Friday, Oct. 19
  Take-home portion, due Friday, Oct 19: P. 114 of Rudin, Exercises 11, 22, 25.
• Midterm Exam Redo due Friday, Oct. 26
  For each part of each of problems 1-3, you can obtain 80% of full credit, provided your writeups are correct, clear, and concise. There will be no partial credit.
• Homework Assignment 8: Due Friday, Oct. 26
  P. 138 of Rudin, Exercises 1,2,4,5.
• Homework Assignment 9: Due Friday, Nov. 2
  Prove parts (a)-(e) of Theorem 6.12.
• Homework Assignment 10: Due Friday, Nov. 9
  P. 138 of Rudin, Exercises 3,7,10.
  Prove the mean value theorem for integrals: If f is continuous and g is nonnegative and Riemann integrable on [a,b], then there exists a point c in [a,b] for which \int_a^b f(x)g(x)dx = f(c) \int_a^b g(x) dx.
• Homework Assignment 11: Due Friday, Nov. 16
  Prove Theorem 7.9 on p. 148 of Rudin.
  P. 165 of Rudin, Exercises 1,2.
• Homework Assignment 12: Due Friday, Nov. 30
  Provide a counterexample to show that the conclusion to Theorem 7.16 is false is f_n -> f pointwise but not uniformly.
  Let f(x) = sum_n c_n sin(nx), x in [0,pi]. Proved nontrivial conditions on the coefficients c_n which guarantee that (a) the series for f(x) converges uniformly on [0,pi]; and (b) the series obtained by termwise differentiation converges uniformly. Confirm convergence by applying the appropriate theorems, and verify that the hypotheses of these theorems do hold.
  P. 165 of Rudin, Exercises 4,5,7,9,16.
  Let G consist of all continuous functions g:[0,1]->R for which |g(x)| is less than or equal to 1 on [0,1], and let F consist of all definite integrals of functions g in G, f(x)=int_0^x g(t) dt. Prove that the family F is compact in the space C[0,1].
• Homework Assignment 13: Due Friday, Dec. 7
  Click here for homework assignment in PDF format.
• Takehome Final: Due Noon Thursday, Dec. 13
  P. 239 of Rudin, Exercises 5 (special case of the Reisz representation theorem), 16, 17a-c, 20, 23. In excercise 17, show by example that f is not globally 1-1.