MATH 4321 Real Analysis

MATH 4321 Real Analysis, Fall 2024

Syllabus

Class meeting

Hall 219 MW 3:30pm – 4:45 pm

Office hours

Hall 314U 4:45 pm– 5:30 pm; Anytime by appointment (please email me)

Main Reference

Typed notes below

Suggested References:

Bartle's Introduction to Real Analysis,
Abbot’s Understanding Analysis,
Laczkovich-Sos' Real Analysis: Foundations and Functions of One Variable,
Pugh's Real Mathematical Analysis,
Rudin's Principle of Mathematical Analysis
"Survey Article: The real numbers–A survey of constructions." Rocky Mountain J. Math. 45 (3) 737 - 762, 2015. https://doi.org/10.1216/RMJ-2015-45-3-737

Typed Notes

Real Analysis: Functions of a real variable (last update: Nov. 19, 2024, 8:07 p.m.)

If(when) you find a typo please email me :)

(Raw) Notes

Sets and Functions (Exercises)

- Sets
- Functions

The Natural numbers and countable sets (Exercises)

- The natural numbers
- Finite Sets
- Countable sets
- Sets Exercise Class

The real numbers (Exercises)

- Fields
- Intervals
- ** Extra credit and Group work **

Sequences and Series of Real numbers (Exercises)

- Sequences
- Properties of sequences
- Series

Topology of R (Exercises)

- Open sets
- Closed and compact sets

- Limits

Continuity (Exercises)

Derivatives (Exercises)

Homeworks

There will be a homework each week. These assessed questions will collectively count for 20% of your overall grade.

HW 1: Choose 5 problems of the exercises sheet for 'Sets and Functions'. Deadline: Sep 4 (Solutions .tex template I use TexShop(local) or overleaf(online) to edit latex files)***SOLUTIONS***
HW 2: Choose 5 problems of the exercises sheet for 'The Natural numbers and countable sets'. Deadline: Sep 11 ***SOLUTIONS***
No Homework this week due to the Exam next week.
HW 3: 1,2,3,4,5 of 'The real numbers' exercises sheet. Deadline: Sep 30
Extra credit 1: Choose one: Explain Dedekind's constructions of the real numbers (Dedekind cuts) or Explain Cantor's construction of real numbers (Cauchy Sequences). Deadline: Sep 30
HW 4: 15,19,22,30,31 of 'The real numbers' exercises sheet. Deadline: Oct 14
HW 5: 1,4,6,12,13 of 'Sequences and Series of Real number' exercises sheet. Deadline: Oct 21 ***SOLUTIONS***
HW 6: 17,21,23,29,30 of 'Sequences and Series of Real number' exercises sheet. Deadline: Oct 23 ***SOLUTIONS***
No homework this week, please study for the group work we began today (namely, the exercise sheet 'Topology of R')
HW 7: 1,4,6,7,8 of 'Continuity' exercises sheet. Deadline: Nov 11

Exams