### Math 321 - Real Variables

Syllabus

Some online textbooks and course notes:

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.