Welcome to the class Introduction to Real Analysis. Real analysis concerns the study of real numbers and functions of a real variable. This includes the study of sequences and series, convergence, continuity of functions, differentiation of functions, integration of functions, and related-areas. Real analysis also serves as the foundation for understanding rigorously a variety of mathematical fields, including calculus, partial differential equations, probability theory, and optimization. This class will cover topics central to the study of functions of a real variable and central to the study of related fields of mathematical analysis.
Selection of Topics
- Functions of Several Variables
- Vector Analysis
- Inverse Function Theorem
- Implicit Function Theorem
- Differential Forms
- Stokes' Theorem
- Gauss Divergence Theorem
- Power Series
- Special Functions: Trigonometric, Exponential, Gamma.
- Fourier Series
- Lebesgue Measure.
- Lebesgue Integral.
- Hilbert Spaces.
- L^p Spaces.
For more information see the syllabus Δ. Please be sure to read the prerequisites and grading policies for the class.
Prerequisites:
Knowledge of how to write rigorous mathematical proofs and a familiarity with epsilon-delta-based proofs.
Grading:
The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final project as follows:
Homework Assignments 30%
Midterm Exam 30%
Final Exam 40%
Homework Policy:
Assignments will be made weekly and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While it is permissible for you to discuss materials with classmates, the submitted homework must be your own work.
Exams:
A midterm exam will be given in the class on Thursday, May 5th.
Final exam will be a take home due on Wednesday, June 8th.
Midterm Outline:
- Power Series.
- Term-wise differentiation and convergence to derivative.
- Behaviour in limit as |x| -> R.
- Form for exponential and logarithm.
- Special Functions.
- Exponential Function.
- Logarithmic Function.
- Gamma Function.
- Fourier Series.
- Relationship between coefficients of the trigonometric expansion and complex exponential expansion.
- Dirichlet kernel.
- Conditions for point-wise convergence.
- Conditions for convergence in norm.
- Parseval's Theorem.
- Linear Transformations
- Properties of finite dimensional vector spaces.
- Matrix representation in terms of a basis.
- Matrix representations of the same transformation when changing the basis.
- Relationship between the kernel and range of linear transformation from X to X.
- Norm of linear transformations.
- Invertibility of linear transformations.
Supplemental Class Notes and Materials:
Class Annoucements:
- No office hours on Tuesday, May 17th.
- Final Exam (Take-Home): [PDF] Δ.
- Fixed typo on problem 3a on the take-home final exam.
- Fixed typo on problem 1bi on the take-home final exam.
- Added hint for computing expression in problem 1bi.
TA and Office Hours:
The TA for the course is Jon Karl Sigurdsson. His office hour is on Wednesday in SH 6431 P from 4pm - 5pm. His Mathlab hours are on Thursdays in SH 1607 from 5pm - 7pm.
Homework Assignments:
Turn all homework into Sigurdsson mailbox in South Hall on the 6th Floor by 4:30pm on the due date. Graded homeworks will be returned in class. Numbered exercises are from Principles of Mathematical Analysis, Walter Rudin, third edition.
HW1: (Due Thursday, April 7th) Ch. 8: 1, 2, 3, 4cd, 6, 9.
HW2: (Due Thursday, April 14th) Ch. 8: 8, 10, 12, 13, 14, 15.
HW3: (Due Friday, May 6th) Ch. 9: 1, 2, 3, 4, 5, 7, 8, 10.
HW4: (Due Thursday, May 12th) Ch. 9: 11, 12abcd, 13, 14, 15.
HW5: (Due Tuesday, May 17th) Ch. 9: 15, 16, 17, 25.
HW6: (Due Tuesday, May 24th) Ch. 9: 27, 30; Ch. 10: 1, 3, 4, 7, 8.
Final Exam (Take-Home): [PDF] Δ (Due Wednesday, June 8th in my mailbox in South Hall 6th floor by 4:30pm).
Edit:
Main |
Menu |
Description |
Info |
Image |
Extras ||
View Extras |
Log-out