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(Term 2, 2010: Jan 4, 2010 -- Apr 15, 2010)

Math 321:201: Real Variables II   (UBC course page is here )

Class: MWF. 9am --10am at MATH 104.
Office hours: MWF 10am -- 11am at MATH 235. Or by appointment. Please email at yhkim "at"  math "dot" ubc 'dot' ca

Text: Walter Rudin, Principles of Mathematical Analysis, third edition.

The Riemann-Stieltjes integral (§6)
Sequences and Series of Functions (§7)
Power Series, Special Functions and Fourier Series (§8)
a supplementary topic like Integration of Differential Forms (§10)

Useful link: Joel Feldman's course page.

Course syllabus


HW #1. Due: Fri. Jan. 15 (hand-in in class). Solutions
HW #2. Due: Fri. Jan. 22 (hand-in in class). Solutions
HW #3. Due: Fri. Jan. 29 (hand-in in class). Solutions
HW #4. Due: Fri. Feb. 5 (hand-in in class).  Solutions
HW #5. Due: Fri. Feb. 12 (hand-in in class). Solutions
HW #6. Due: Monday, March 1. (hand-in in class). Problem 4 was modified for more clear statements. Solutions.
Midterm Guideline.
HW #7. Due: Fri. March 19 (hand-in in class). Solutions
Reading assignment: (due: Sunday. March 21) (self reading: [Rudin, p 204 --p218]. Especially [Rudin 9.7, 9.8, 9.11, 9.12, 9.13, 9.15, 9.17, 9.18, 9.19].)
HW #8. Due: Fri. March 26 (hand-in in class). There was a typo in Problem 4 (b). It is now corrected: the condition should read "f(a)=f(b) = 0".  Solutions
HW #9. Due: Wed. April 7 (hand-in in class).
HW #10. NOT to be handed-in.


Jan. 11, 2010.
The official dates for the April exam schedule are April 19th to May 1st.  The lateness of the schedule is due to the two week class suspension to accommodate the Olympics.  The students must wait until the exam schedule is released (mid-February) and they are certain of their exam dates before making any plans.  Students who ignore this warning will not be accommodated by the Math Department with alternate exam sittings.  The usual accommodations due to conflicts or hardships (as defined by UBC) will be assessed on a case by case basis.  There is also a possibility that there will be exams scheduled on Sunday, April 25th, in order to accommodate every course.

Jan. 13, 2010.

Are you a citizen or permanent resident of Canada?
Then you may want to consider NSERC Undergraduate Summer Research Awards (USRA):

Jan. 24, 2010. There was a typo in HW #3, Problem 5 regarding the definition of upper integral. It is now fixed.

Jan. 27, 2010.  Now we have additional office hours (run by Mr. Simon Rose) on Thursdays at LSK 202B for  9--11am.

Feb. 7, 2010. There was a typo in HW#5 Problem 1. It is now corrected.

Feb. 11, 2010. Final exam is now scheduled on April 20, 2010 at 8:30 am. (Location will be decided later.)

Feb. 11, 2010. In HW #6, Problem 4 was modified for more clear statements.

Feb. 12, 2010. Midterm exam will be on Friday, March 5, in class. It is important for you to look at this Midterm Guideline.

Mar. 15, 2010. There is a undergraduate colloquium series ( ) in UBC. Your attendance is highly recommended!

Schedule / Plan / Progress / Summary

Week Date Contents
1 0104 Rudin Ch. 6. Definition of Riemann-Stieltjes integral. Example of non Riemann integrable function. Dirac-delta function (or integral with resptect to a step function).
0106 Integrability criteria: Upper partial sum - lower partial sum. Continuous functions are integrable.
0108 We have covered Rudin up to 6.10 + alpha. Integralibity of piece-wise continuous functions.  (HW #1 posted)
2 0111 Composition of functions and integrability. Linearity. Products and integrability.
0113 Change of variable. Fundamental theorem of calculus.
0115 Reduction to Riemann integral. Fundamental theorem of calculus (a second version). Integration by parts. (HW #1 due in class), (HW#2 posted).
3 0118 Integration of vector-valued functions. Rectifiable curves. (Recall that there is NO Mean Value Theorem for vector valued functions: find a counterexample.)  (HW#1 solution posted)
0120 Ch.  7. Sequences and Series of Functions. convergence of functions. uniform convergence.
0122 Uniform convergence and operations on functions: uniform convergence and limit of functions and continuity (Rudin 7.11, 7.12). (HW#2 due in class. HW#3 is posted)
4 0125 uniform convergence and limit of functions and continuity (continued). Convergence and integral.
0127 Convergence and integral: examples of relation between pointwise convergence and integral. uniform convergence and integral (Rudin 7.16).  (application. see also Rudin 7.10). (proof of theorem).
0129  Convergence and differentiation (Rudin 7.17).  nowhere differentiable continuous functions (Rudin 7.18)    (HW#3 is due in class, HW#4 is posted)
5 0201 compactness in the space of functions I: space of continuous functions. completeness of C(X, Y) when Y is a complete metric space (Rudin 7.15).
0203 compactness in the space of functions II: pointwised boundedness. uniform boundedness. equicontinuity. Arzela-Ascoli theorem (Rudin 7.19 -- 25): proof
0205 compactness in the space of functions III:Arzela-Ascoli theorem: proof. 
Stone-Weierstrass theorem I (Polynomial approximation (Rudin 7.26))   (HW#4 is due in class, HW#5 is posted)
6 0208 Stone-Weierstrass theorem (Rudin 7.26) II. Convolutions of two functions.
0210 Stone-Weierstrass theorem III: the proof of Weierstrass theorem (Rudin 7.26). Stone's generalization (Rudin 7.32).
0212 Stone-Weierstrass theorem IV: Stone's generalization: density of a lattice (Rudin 7.32, steps 3, 4).   (HW#5 is due in class, HW#6 is posted)
0214-0227 Midterm break
0301 Stone-Weierstrass theorem V: Stone's generalization: density of an algebra. (Rudin 7.28 --- 33)
 (HW #6 is due in class)
0303 Stone's theorem: complex valued version. Ch. 8. Special functions. Fourier series I: a motivation (Rudin 8.9).
0305 Midterm Exam in Class
0308 Fourier series II: relation to differential equations. Fourier coefficient. Examples. 
0310 Fourier series III: Fourier coefficient: examples, linearity, elementary estimates, derivatives
0312 Fourier series IV: convergence of Fourier series to the original function (when?). Parseval's theorem (without proof). uniform convergence of Fourier series (when?).   (HW #7 is posted)
0315 Fourier series V : convolution. Dirichlet kernel [Rudin 8.13]. Pointwise convergence of Fourier series (when?).  localization of convergence [Rudin 8.14 Corollary].  (HW #8 is posted)
0317 Fourier series VI : proof of pointwise convergence of Fourier series [Rudin 8.14]. Riemann-Lebesgue lemma [Rudin 8.12].  L^2 theory (orthogonality) [Rudin 8.10, 8.11].
0319 Fourier series VII: L^2 theory [Rudin 8.11, 8.12]. L^2 space of functions. Orthogonality. Pythagoras theorem. Orthogonal projection.   (HW #7 is due in class)
12 0322 (Reading assignment due Sun. March 21). Parseval's theorem [Rudin 8.16]. Proof
0324 Ch. 9. Mappings from R^n to R^m. (self reading: [Rudin, p 204 --p218]). Derivative: the meaning of its definition [Rudin 9.11 --]
0326 Continous differentiabiltiy [Rudin 9.21] (HW #8 is due in class.)
13 0329 Continous differentiabiltiy [Rudin 9.21], Contraction principle [Rudin, 9.23].
0331 Contraction principle [Rudin 9.23]. Inverse function theorem [Rudin, 9.24 -25]. 
0402 Good Friday
14 0405 Easter Monday
0407 Inverse function theorem. Implicit funciton theorem [Rudin 9.26 --29]. (HW #9 is due in class.) (HW#10 is posted)
0409 Implicit function theorem.
15 0412 Partitions of Unity (Rudin 10.8). 
0414 Definition of Manifold
0416 no class
April 20
Final Exam. at 8:30am