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.
Topics:
The RiemannStieltjes 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.
Assignments
HW #1.
Due: Fri. Jan. 15 (handin in class). Solutions
HW #2. Due: Fri. Jan. 22 (handin in class).
Solutions
HW #3. Due: Fri. Jan. 29 (handin in class). Solutions
HW #4. Due: Fri. Feb. 5 (handin in
class). Solutions
HW #5. Due: Fri. Feb. 12 (handin in class). Solutions
HW #6. Due: Monday, March
1. (handin in class). Problem 4 was modified for more clear
statements. Solutions.
Midterm Guideline.
HW #7. Due: Fri. March 19 (handin 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 (handin 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 (handin in class).
HW #10. NOT to be handedin.
Week  Date  Contents 
1  0104  Rudin Ch. 6. Definition of RiemannStieltjes integral.
Example of non Riemann integrable function. Diracdelta 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
piecewise 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 vectorvalued 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. ArzelaAscoli theorem (Rudin 7.19  25): proof  
0205  compactness in
the space of functions III:ArzelaAscoli theorem: proof. StoneWeierstrass theorem I (Polynomial approximation (Rudin 7.26)) (HW#4 is due in class, HW#5 is posted) 

6  0208  StoneWeierstrass theorem (Rudin 7.26) II. Convolutions of
two functions. 
0210  StoneWeierstrass theorem III: the proof of Weierstrass
theorem (Rudin 7.26). Stone's generalization (Rudin 7.32). 

0212  StoneWeierstrass 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) 

02140227  Midterm break  
9 
0301  StoneWeierstrass 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 

10 
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) 

11 
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]. RiemannLebesgue 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 