Mathematics 320

Real Variables I

Department of Mathematics,  University of British Columbia

Prerequisites: Either (a) a score of 68% or higher in MATH 226 or (b) one of MATH 200, MATH 217, MATH 226, MATH 253, MATH 263 and a score of 80% or higher in MATH 220.

Instructor

Joel Feldman

Text

• Walter Rudin, Principles of Mathematical Analysis, third edition.

I will post all handouts, problem sets, etc. on the web here.

Other References

• Tom M. Apostol, Mathematical Analysis.
• Maxwell Rosenlicht, Introduction to Analysis.
• Arthur Mattuck, Introduction to Analysis.
• Kenneth Ross, Elementary Analysis: The Theory of Calculus
• William F. Trench, Introduction to real analysis

Topics

1. Number Systems (§1):
       ordered fields
       rational, real and complex numbers
       Archimedian property
       supremum, infimum, completeness
2. Sequences and Series of Real Numbers (§3):
       limits of sequences, algebra of limits
       Bolzano-Weierstrass Theorem
       Cauchy sequences, liminf, limsup
       limits of series, convergence tests, absolute and conditional convergence
       power series
3. Metric Spaces (§2):
       metric spaces
       convergence, completeness, completion
       open sets, closed sets, compact sets, Heine Borel Theorem
       connected sets
4. Continuity (§4):
       functions, cardinality
       continuity
       continuity and compactness, existence of minimizers and maximizers, uniform continuity
       continuity and connectedness, Intermediate Value Theorem
       monotone functions and discontinuities
5. Differentiation (§5):
       differentiation
       Mean Value Theorem
       L'Hôpital's Rule
       Taylor's Theorem

Grading

• There will be two midterms (on Wednesday, October 1 and Wednesday, November 5), each accounting for about 20% of the final mark.
• There will be weekly problem sets accounting for about 10% of the final mark.
• The final exam will account for about 50% of the final mark.
• Grades will probably be scaled.

Policies

• The midterm and final examination will be strictly closed book: no formula sheets or calculators will be allowed.
• Late homework assignments receive a grade of 0.
• Missing a homework or midterm normally results in a mark of 0. Exceptions may be granted in two cases: prior consent of the instructor or a medical emergency. In the latter case, the instructor must be notified within 48 hours of the missed test, and presented with a doctor's note immediately upon the student's return to UBC. When an exception is granted for a missed midterm, there is no make-up midterm, and the final exam mark will be used instead.