This schedule is tentative. It will get more accurate as the semester
progresses. No week's topics should be taken as final until the homework
is linked.
Week #1:
Aug 30 - Sep. 3
Homework
|
Metric Spaces
Compact Sets
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Finite, countable infinite and uncountable sets.
NxN and Q are countable, binary operations.
Groups, Examples, identity and inverse elements.
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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 Vector-valued Functions
Sequences and Series of Functions: Discussion of Main Problem
|
|
Administrative note: Friday 22 October is the Mid-Semester 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 L2
|
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Administrative note: Friday 10 December is the last day of class.
|
Final Exam
Dec 10 - 21
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The Final Exam
will be a take some exam.
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