MATH 400 Real Analysis
Instructor: Hemanshu Kaul
Office: 125C, Rettaliata Engg Center.
E-mail: kaul [at] iit.edu
Class Time: 3:15-4:30pm, Tuesday and Thursday
Place: 122, Rettaliata Engg Center
Discussion Forums: Math 400 at Campuswire.
Office Hours: Tuesday and Thursday at 1:45-2:45pm. And by appointment in-person or through Zoom (send email to setup appointment).
Questions through Campuswire Discussion Forum are strongly encouraged.
TA Office Hours: Jared Roush. Wednesday 12pm-3pm at RE 129 or through Zoom link at Math Tutoring Center. You can also talk to Ziheng Guo at RE 129 who has been a TA for Math 400 in the past.
|Course Information|
|Advice|
|Announcements|
|Examinations|
|Weekly Class Log & HW|
|Links|
Course Information:
This course is an introduction to the rigorous foundations of single-variable calculus - real number system; limits; convergence of sequences and series; continuity, differentiability, and integrability of functions.
The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, as well as other relevant information. Read it carefully!
What is this course really about? Required reading.
The end-of-semester letter for students: What Next?
The official course syllabi: MATH 400 (reference for topics only).
Advice for students:
Excellent advice by Francis Su on good mathematical writing.
Why do we have to learn proofs?
Understanding Mathematics - a study guide
On a more abstract note, here is a discussion by Tim Gowers on Language and Grammar of Mathematics - which is what you are learning in a course like this.
Excellent advice for math majors, especially those planning to go on to graduate school, by Terry Tao, 2006 Fields medallist. Required reading.
Read this book on a variety of experiences in the journey to learn mathematics:
Living Proof
Some of the primary sources of information/discussion for careers in Mathematical Sciences:
MAA - Careers
SIAM - Careers
INFORMS - Careers
AMS - Careers
Class Announcements:
- Tuesday, 8/22 : Check this webpage regularly for weekly lecture topics, videos, and HW.
Examinations:
- Exam # 1 : Thursday, October 5th. Syllabus: Based on topics corresponding to HWs #1 to #5.
- Exam # 2 : Thursday, November 16th. Syllabus: Based on topics corresponding to HWs #6 to #10.
- Final Exam : Thursday, December 7, 10:30am – 12:30pm, RE 122. Topics: All topics studied during the semester.
Weekly Class Log with Videos, Discussion Questions, Notes, and HW:
- Week #1 : 2 lectures
- Topics and Readings: Examples for limitations of Calculus and other dangers, Irrational Numbers, Basic properties of Real Numbers, Axiom of Completeness, Supremum and Infimum, Maximum and Minimum. From: Sections 1.1, 1.2, 1.3, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account). Videos #1-#4.
- Discussion and Review Questions: Discussion/ Review Questions #1 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#1.
- Homework & Comments: Homework 1. Due 10pm, August 31st. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #2 : 2 lectures
- Topics and Readings: Nested Interval Property, Q is dense in R, Existence of Roots, Cardinality and (Un)Countable sets, Cantor's Diagonalization Method, Cantor's Theorem and Infinity of Infinities. From: Sections 1.4, 1.5, 1.6, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #5-#8.
- Discussion and Review Questions: Discussion/ Review Questions #2 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#2.
- Homework & Comments: Homework 2. Due 10pm, September 7th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #3 : 2 lectures
- Topics and Readings: Sequences - Convergence, Uniqueness of limit, Divergence, Algebraic and order properties of limits. From: Sections 2.2, 2.3, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #9-#11.
- Discussion and Review Questions: Discussion/ Review Questions #3 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#3.
- Homework & Comments: Homework 3. Due 10pm, September 14th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #4 : 2 lectures
- Topics and Readings: Monotone Convergence Theorem, Convergence and Divergence of Series, Subsequences and Bolzano-Weierstrass Theorem. From: Sections 2.4, 2.5, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #12-#14.
- Discussion and Review Questions: Discussion/ Review Questions #4 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#4.
- Homework & Comments: Homework 4. Due 10pm, September 21st. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #5 : 2 lectures
- Topics and Readings: Cauchy sequences and Cauchy criterion for convergence of sequences, Algebra of Series Limits, Cauchy criterion for Series, Comparison Test for series, Series p-test, Geometric series, Absolute convergence test, Alternating Series test, Absolute and Conditional convergence of series, Rearrangements of Series - absolutely convergent series vs conditionally convergent series. From: Sections 2.6, 2.7, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #15-#17.
- Discussion and Review Questions: Discussion/ Review Questions #5 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#5.
- Homework & Comments: Homework 5. Due 10pm, September 28th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Weeks #6 and #7 : 3 lectures and 1 Mid-term Exam
- Topics and Readings: The Cantor set and its properties, Open set and properties, Limit point of a set, Closed set and properties, Closure of a set, Complements of open sets and closed sets, Compact sets. From: Sections 3.1, 3.2, 3.3, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #18-#20.
- Discussion and Review Questions: Discussion/ Review Questions #6 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#6.
- Homework & Comments: Homework 6. Due 10pm, October 12th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #8 : 2 lectures
- Topics and Readings: Discussion of Exam#1. Sequential Compactness, Compactness (Every open cover has a finite subcover), Closed and Bounded sets, Heine-Borel Theorem, Limit of a function, Sequential Characterization of Functional limits, Algebra of Functional limits. From: Sections 3.3, 4.2, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #21-#22.
- Discussion and Review Questions: Discussion/ Review Questions #7 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#7.
- Homework & Comments: Homework 7. Due 10pm, October 19th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #9 : 2 lectures
- Topics and Readings: Discussion of Exam#1. Proofs related to Sequential Characterization of Functional limits; Continuity, Examples, Preservation of Compactness by continuous functions, Extreme Value Theorem, Bolzano's Theorem for existence of solutions, Intermediate Value Theorem. From: Sections 4.1, 4.2, 4.3, 4.4, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #23-#25.
- Discussion and Review Questions: Discussion/ Review Questions #8 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#8.
- Homework & Comments: Homework 8. Due 10pm, October 26th. Submit a PDF file through Blackboard Assignment. Solutions distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #10 : 2 lectures
- Topics and Readings: Uniform Continuity - examples, nonexamples, compact domains; Differentiability, Algebra and Chain rule, Interior Extremum by Derivative, Darboux' Theorem, Unusual examples/ Topologist's sine curve. From: Sections 4.4, 5.2, 5.1, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #26-#28.
- Discussion and Review Questions: Discussion/ Review Questions #9 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#9.
- Homework & Comments: Homework 9. Due 10pm, November 2nd. Submit a PDF file through Blackboard Assignment. Solutions to be distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Week #11 : 2 lectures
- Topics and Readings: Darboux' Theorem. Rolle's Theorem, Mean Value Theorem, Generalized (Cauchy's) MVT, Consequences of MVT, L'Hospital's Rules, Pointwise and Uniform Convergence of sequence of functions, Preservation (?) of continuity and differentiability under convergence of functions. From: Sections 5.3, 6.2, 6.3, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #29-#30.
- Discussion and Review Questions: Discussion/ Review Questions #10 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#10.
- Homework & Comments: Homework 10. Due 10pm, November 9th. Submit a PDF file through Blackboard Assignment. Solutions to be distributed in class.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Weeks #12 & #13 : 3 lectures and 1 Mid-term Exam
- Topics and Readings: Series of functions and its pointwise and uniform convergence, Power series, its radius of convergence, pointwise convergence and uniform convergence, Properties of Power series - continuity, term-by-term differentiation and antidifferentiation, Taylor (McLaurin) Series, Error function and Lagrange Remainder theorem; Riemann Integral, Upper and Lower Integrals, Examples and non-examples, Integrability Criterion, Continuous Functions are integrable, Integrating functions with finitely many points of discontinuity. From: Sections 6.4, 6.5, 6.6, 7.2, 7.3, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #31-#34.
- Discussion and Review Questions: Discussion/ Review Questions #11 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#11.
- Homework & Comments: Homework 11. Due 11:59pm, November 22nd (Wednesday). Submit a PDF file through Blackboard Assignment.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
- Weeks #14 & #15 : 3 lectures and 1 holiday
- Topics and Readings: Continuous Functions are integrable, Integrating functions with finitely many points of discontinuity, Examples of Integrating functions with infinitely many points of discontinuity, Discussion of Lebesgue's criterion for Riemann Integrability (without proof); (More) Basic Properties of Integrable functions - algebra, comparison, Integral version of Triangle Inequality; Interchange of limit and integral under uniform convergence; Integral Mean Value Theorem; Fundamental Theorem of Calculus, Application of FTOC to prove Integration by Parts and Substitution Rule. From: Sections 7.4, 7.5, and elsewhere.
- Pre-recorded Lecture Videos: Available at YouTube (log in through your IIT account).
Videos #35-#36.
- Discussion and Review Questions: Discussion/ Review Questions #12 based on this week's lectures.
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#12.
- Homework & Comments: Homework 12. Due 10am, December 1st (Friday). NOTE THE SPECIAL TIME and INSTRUCTIONS. Submit a PDF file through Blackboard Assignment.
Ask for help through Campuswire Discussion Forums, during the instructor and TA office hours, or through email to instructor.
Links for Additional Information:
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