Syllabus

Course notes by Adrian She

Some online textbooks and course notes:

- An introduction to real analysis by William Trench.
- Analysis
Webnotes by John Lindsey Orr.

- MIT open courseware.
- Introduction to Real Analysis by Lee Larson.

**Links to some more advanced topics:
**

- Riesz Proves the Riesz Representation Theorem by Mike Bertrand. Also has translations of the original articles by Riesz.
- An everywhere continuous nowhere differentiable function by John McCarthy.
- Some
informal notes about Lp-spaces and convergence by Harald
E. Krogstad. (The Lp-spaces are properly defined using Lebesgue
measure and Lebesgue integral, but the main ideas can be
understood using Riemann integral. The abbreviation a.e. stands
for almost everywhere, meaning everywhere except on a set of
measure zero. For a measure space with finite measure you can
take an interval [a,b], and for a measure space with infinite
measure the real line.)

Homeworks:

Homework #1 Solutions

Homework #2 Solutions

Homework #3 Solutions

Homework #4 Solutions

Practice Exam #1 Solutions

Exam #1 Solutions

Homework #5 Solutions

Homework #6 Solutions

Homework #7 Solutions

Homework #8 Solutions

Homework #9 Solutions

Practice Exam #2 Solutions

Exam #2 Solutons

Homework #10 Solutions

Topics covered:

- Week 1: Jan 4 - Jan 8
- Definition of Riemann-Stieltjes integral. Examples with step
functions. 6.1-6.2, 6.14-6.16

- Week 2: Jan 11 - Jan 15
- Criterion for integrability, continuous functions, finite
number of discontinuities, monotone functions, composition of
functions. Riemann sums. 6.3-6.11.

- Week 3: Jan 18 - Jan 22
- Properties of the integral. The case of differentiable
alpha. Change of variable formula. The fundamental theorem of
calculus. 6.17-6.22.

- Week 4: Jan 25 - Jan 29
- Rectifiable curves, length of curves. Sequences and series of functions. Convergence and uniform convergence. 6.26, 7.1-7.10.
- Week 5: Feb 1 - Feb 5
- Continuity and uniform convergence. Integration and uniform convergence. 7.11-7.16.
- Week 6: Feb 8 - Feb 12
- Exam #1. Differentiation and uniform convergence. 7.17.
- Week 7: Feb 22 - Feb 26
- Equicontinuous sequences and the Arzela-Ascoli theorem. 7.19-7.25.
- Week 8: Feb 29 - Mar 4
- Peano's theorem, Weierstrass approximation theorem. 7.26-7.27.
- Week 9: Mar 7 - Mar 11
- Stone-Weierstrass theorem, the lattice version and the algebra version. 7.28-7.33.
- Week 9: Mar 14 - Mar 18
- Power series. 8.1-8.5.
- Week 10-12: Mar 21 - Apr 8
- Fourier series. 8.9-8.16.