Lectures Notes: Introduction to Real Analysis

Syllabus for MA504

 Midterm 1: Wednesday October 4, 8:00-10:00 p.m. Location: SMTH 118

 Miderm 2: Wednesday November 15, 8:00-10:00 p.m. Location: SMTH 118

 Office Hours: Wednesday 1:30-2:30 p.m. and Thursday (by appointment)

Lecture 1: Real numbers

Lecture 2: Supremum and Infimum

Lecture 3: Euclidean spaces, Metric spaces, Functions

Lecture 4: Finite Sets, Countable sets, Uncountable sets

Lecture 5: Open sets, closed set, limit points

Lecture 6: The closure of a set

Lecture 7: Compact sets

Lecture 8: Heine-Borel Theorem

Lecture 9: Perfect sets, Cantor set

Lecture 10: Connected Sets

Lecture 11: Convergence of sequences

Lecture 12: Bolzano-Weierstrass theorem, liminf and limsup of a sequence

Lecture 13: Cauchy sequences, Complete metric spaces

Lecture 14: Limits of functions, Continuous functions

Lecture 15: Continuous functions map compact sets to compact sets, and connected sets to connected sets

Lecture 16: Uniformly continuous functions, discontinuities of the first and second kind

Lecture 17: Monotonic functions, Inverse function

Lecture 18: Differentiability of functions of one variable, chain rule, mean value theorem

Lecture 19: L'Hospital Rule, Taylor Theorem

Lecture 20: Riemann integral, Integrability Criteria

Lecture 21: Continuous functions are Riemann integrable, Monotonic functions are Riemann integrable, Properties of the integral

Lecture 22: Fundamental Theorem of Calculus

Lecture 23: Change of Variables, Length of a curve

Lecture 24: Sequences of functions, Pointwise covergence, Uniform convergence

Lecture 25: The uniform limit of continuous functions is continuous, the spaces of countinuous and bounded functions is a complete metric space

Lecture 26: Uniform convergence and integration, uniform convergence and derivation, a function that is nowhere differentiable

Lecture 27: Equicontinuity

Lecture 28: Arzela-Ascoli Theorem: Gives the existence of a uniformly convergent subsequence

Lecture 29: Stone-Weierstrass Theorem: A continuous function on an interval can be approximated uniformly by polynomials

Lecture 30: A usuful corollary of Stone-Weierstrass Theorem; The space of polynomials in dense in C([a,b])

Lecture 31: Stone's generalization of Stone-Weierstrass Theorem

Lecture 32: Proof of Stone's Theorem

Lecture 33: Functions of several variables, differentiability

Lecture 34: Inverse Function Theorem

Lecture 35: Introduction to Lebesgue Theory of Integration

  Homeworks

Homework 1 (Due date: Sunday September 3)

Homework 2, Chapter 2 (Sunday September 10): 12, 14, 15, 16

Homework 3 (Sunday September 17)

Homework 4 (Sunday September 24), Chapter 3: 1, 2, 5.

Homework 5 (Sunday October 1), Chapter 3: 20, 21, 23.

Homework 6 (Sunday October 15), Chapter 4: 1, 2, 3, 4, 7, 18, 20, 21.

Homework 7 (Sunday October 22), Chapter 5: 1, 2, 5, 26, 27

Homework 8 (Sunday October 29), Chapter 6: 1, 2, 5, 6, 11, 12

Homework 9 (Sunday November 5), Chapter 7: 1, 2

Homework 10 (Sunday November 12), Chapter 7: 5, 8, 10

Homework 11 (Sunday December 3), Chapter 7: 16, 18, 19, 20, 22