Week #1:
Aug 30  Sep. 3
Homework

Metric Spaces
Compact Sets

Finite, countable infinite and uncountable sets.
NxN and Q are countable, binary operations.
Groups, Examples, identity and inverse elements.

Week #2:
Sep. 6  10
Homework
 Compact Sets
Connected Sets
 Rings, Examples
Fields, Examples, the Real numbers

Administrative Note: Monday 6 September is Labor Day.
Class will not meet.

Week #3:
Sep 13  17
Homework
Problem Session 1

Convergent Sequences
Subsequences
Cauchy Sequences

Homomorphisms of groups, normal subgroups.

Week #4:
Sep. 20  24
Homework
Problem Session 2

Series
Series of Nonnegative terms
The Root and Ratio Tests
Power Series
Summation by Parts

Quotient groups, the First Isomorphism Theorem.

Week #5:
Sep. 27  Oct. 1
Homework
Problem Session 3

Absolute Convergence
Addition and Multiplication of Series
Rearrangements
Limits of Functions
Continuous Functions


Week #6:
Oct. 4  8
Homework
Problem Session 4

Continuity and Compactness
Continuity and Connectedness
Discontinuities
Monotonic Functions
Infinite Limits and Linits at Infinity
Connected, Path Connected and Simply Connected.
Homotopies and the Fundamental Group.


Week #7:
Oct 11  15
Midterm
Problem
Session 5

The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
L'Hospital's Rule
Derivatives of Higher Order


Week #8:
Oct 18  22
Homework
Problem Session 6

Taylor's Theorem
Differentiation of Vectorvalued Functions
Sequences and Series of Functions: Discussion of Main Problem


Administrative note: Friday 22 October is the MidSemester Break.
Class will not meet.

Week #9:
Oct 25  29
Homework

Uniform Convergence
Uniform Convergence and Contuinity
Uniform Convergence and Differentiation

Week #10:
Nov 1  5
Homework
Problem Session 8

Equicontinuous Families of Functions
Definition and Existence of the (Riemann) Integral
Properties of the Integral

Week #11:
Nov 8  12
Homework

Integration and Differentiation
Integration of Vector Valued Functions
Rectifiable Curves

Week #12:
Nov 15  19
Homework
Problem Session 10

Set Functions
Construction of the Lebesgue Measure

Week #13:
Nov 22  26
Homework

Measure Spaces

Week #14:
Nov 29  Dec 3
Homework

Measurable Functions
Simple Functions
Integration

Week #15:
Dec 6  10
Homework

Comparison with the Riemann Integral
Integration of Complex Functions
Functions of Class L^{2}


Administrative note: Friday 10 December is the last day of class.

Final Exam
Dec 10  21

The Final Exam
will be a take some exam.
