II (UBC course page is here
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
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)
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.
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.
|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
|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|
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
|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)
||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
Fourier series I: a motivation (Rudin 8.9).
||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
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
|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].|
|0407||Inverse function theorem. Implicit funciton
theorem [Rudin 9.26 --29]. (HW #9 is
in class.) (HW#10 is posted)
|0409||Implicit function theorem.|
|15||0412||Partitions of Unity (Rudin 10.8).|
||Final Exam. at 8:30am