18798 Math 510 Intro to Analysis I     

Lectures: Dimiter Vassilev, T R 1100-1215    HUM 428

TA/Grader: Justin Pati,

 

                                                                  

Instructor: Dimiter Vassilev     Office :  Humanities Bldg, Office 447  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: TTh 10:00-11:00,  13:30-14:00  or by appointment.

 

Final Exam: Tuesday, December 16, 12:30�2:30 p.m.

 

Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, November 14, 2008. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before November 14, 2008, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses.        

 

 

Text: W. Rudin, Principles of Mathematical  Analysis.

 

Please note the following guidelines for the course:

 

Course content: This is a first graduate course on Analysis. It is also the course that prepares graduate students for the Real Analysis Qualifying. Students should have at this point enough computational background (calculus of one variable and multivariable calculus -at least 2 and 3 variables), and I expect most of you to have been exposed at least once to rigorous epsilon-delta proofs, if not you might consider taking Math 401/501 instead. I expect familiarity with the real and complex numbers.


Description: Real number fields, sets and mappings. Basic point set topology, sequences, series, convergence issues. Continuous functions, differentiation, Riemann integral. General topology and applications: Weierstrass and Stone-Weierstrass approximation theorems, elements of Fourier Analysis (time permitting).

 

Prerequisite: Math 321 and 401.

 

Grades: The final grade will be determined by homework (25%), two midterms( 50%) and a final exam (25%). 

 

Homework:  Homework is due every Wednesday. There will be one weekly homework. You can work together on the homework, but you do need to write up your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly (no points for work that the reader cannot follow- this is true for exams), and staple the sheets.   The two lowest homework grades will be dropped. Please no late homework!  Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the course you will have built a folder with solutions to most of those problems for future reference.

 

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given. 

 

Disability Statement: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs and they should also contact Accessibility Services in Mesa Vista Hall, Room 2021, phone 277-3506.  In addition, they should see CATS- Counseling and Therapy Services; Student Health Center (277-4537). (They can help if you suffer from exam anxiety).    

 

 

                                             Math 510 Syllabus  Fall  2008  (please check after class as advanced postings of homework could change)

 Week of

Topics Covered

Due Homework - turn in Thursday following week. Work on Past Analysis Qualifying Exams

Aug. 25

Chapter 1: The Real and Complex Number Systems

  • Ordered Sets
  • Fields
  • The Real Field

 
  • Ex. For Ch 1, p. 22/ 1, 2, 4, 5.

Sep. 1
  • The Real Field

 
  • Show that the positive real numbers (cuts) are an abelian group w.r.t. the multiplication we defined
  • Ex. For Ch 1: 8

Sep. 8
  • The Complex Field
  • Euclidean Spaces  

 

Chapter 2: Basic Topology

  • Basic Topology
  • Finite, Countable, and Uncountable Sets
  • Metric Spaces

 
  • Prove Theorem 1.20 using the least upper bound property and the fact that the real numbers are a field.
  • Ex. Ch 1, p. 22/ 6
  • Ex. Ch 1, p. 22/ 12, 13
  • Ex. Ch.2, p.43/ 2, 3, 10 (only metric space property), 11

Sep. 15
  • neighborhoods, limit points, closure of a set, interior of set, closed/open sets, perfect sets; unions of open sets (intersections of closed sets; finite intersections of open sets ( finite unions of closed sets).
  • compact sets

 
  • Ex. Ch.2, p.43/ 8, 9, 10 (remaining parts), 12, 13, 19, 15.
  • Past Analysis Quals: p.2 #1 (Fall 2007), p.4 #1 (Spring 2007).

 

Past Analysis Qualifying Exams

Sep. 22

 

  • compact sets
  • perfect sets
  • Ex. Ch.2, p.43/ 18, 25, 26, 30.
  • Quals p.44 #5 (Fall 1995)

Sep. 29
  • perfect sets
  • connected sets

 

Chapter 3: Numerical sequences and series

  • convergent sequences, liminf and limsup
  • Ex. Ch.2, p.43/ 28, 29.
  • Ex. Ch. 3, p 78/1, 2, 3.
  • Past Analysis Quals: p.4 #2, p.31#1, p. 31#2

Oct. 6
  • Cauchy sequences
  • Note (for p.43. ex 18): Try to construct such a set using the set E  in ex. 17 –YES you can.  Remove more than just the rationals; order the rationals in a sequence and then inductively remove a suitable open interval with irrational ends from E (so that these ends are never removed).
  • Ex. Ch. 3, p.78/ 20, 21, 24, 25.
  • Past Analysis Quals: p.22 #3, p.44 #5 c) and d) (Hint: can you cover a point which is on the boundary of some open rectangle used in the cover; using closed rectangles might make a difference – can you do it with sides parallel to the axes? Does the intersection/union of two such rectangles have the wanted cover?)

Oct. 13

October 16-17 Fall Break
  • Monotone sequences; some special sequences
  • Ch. 3, p.81/ 23  

Oct. 20

Exam #1 on Thursday, October 23

 

Oct. 27
  • some special sequences
  • series and power series
  • Ch. 3, pp. 78-82/ 6, 8 (Abel’s test), 9, 17.
  • Past Analysis Quals: p.11 #5, p. 25 # 7, p. 27 #5, p. 34 #1, p. 36 #2.
  • Two problems assigned in class.

 

Nov. 3

 

Chapter 4: Continuity
  • Ch. 4, pp. 98-102/ 1, 2, 3, 4, 5, 6, 8, 11, 13 (also give an example that continuity alone is not enough), 14, 15, 20, 21.
  • Past Analysis Quals: p. 8 #1b, p. 34/ #3, p.10 #1.
  • Car talk puzzler 2006 #23, cartalk.com-puzzler

Nov. 10

Chapter 5: Differentiation
  • Ch. 4, p. 101/ 22   Ch. 5, pp. 114 – 115/ 21, 3, 7, 11, 14, 16, 20, 25
  • Past Analysis Quals: p. 38 #3, p. 2 #3 (see also Rudin, p.115/15),

Nov. 17

Chpater 6: The R-S integral

 

Exam #2
  • Ch. 6, pp. 138-142/ 1, 2, 3, 4, 5, 10, 17 (Due Thursday, December 4)

Nov. 24

November 27-30 Thanksgiving

 

 

Dec. 1

Chapter 7: Sequences and series of functions
  • Ch. 6, pp. 138-142/ 8, 19
  • Past Analysis Quals: p.10 #2, p.12 #6 , p.22 #4, p.34 #2 (Due Thursday, December 11)
  • Ch. 7, pp.165-168/ 4, 6, 9,10

Dec. 8

Chapter 8: Some special functions

·       Ch. 7, pp.165-168/  16, 19

Dec. 15

Final Exam: Tuesday, December 16, 12:30-2:30 p.m.

 

 

 

 

Labor Day Holiday (Saturday classes meet as scheduled) (no classes/University closed) . September 1, 2008

Fall Break (Saturday classes meet as scheduled) (no classes). October 16-17, 2008

Thanksgiving Holiday (no classes/University closed). November 27-30, 2008

Final examinations December 15-20, 2008