UC Davis Math 201B
Analysis
Basic information
| CRN code: | 30184
|
| Room/Time: | MWF 09:00AM-09:50AM in Giedt 1006 |
| Instructor: | Alexander Soshnikov |
| Office: | MSB 3140 |
| Office Hours: | M 11:00-11:50am, plus by appointment
|
| E-mail: |
soshniko@math.ucdavis.edu |
| |
| TAs: | Indrajit Jana |
| TA Office: | MSB 3125 |
| Discussion: | T 09:00AM-09:50AM in Giedt 1006 |
| TA Office Hours: | T 03:10-04:00PM and R 05:10-06:00PM |
| E-mail: |
ijana@math.ucdavis.edu
|
| |
| Midterm (in class): | Friday, February 12, 09:00-10:00AM | |
| Final Exam (in class): | Tuesday, March 15th, 03:30PM-05:30PM
|
| Grading: |
Problem sets 20%, Midterm 40%, Final Exam 40% |
| Webpage: | www.math.ucdavis.edu/~soshniko/201b | |
| |
Topical Outline:
1. Basic Measure and Integration Theory:
Fundamental Definitions from Measure Theory.
Measurable functions and approximation by simple functions.
Dominated and monotone convergence theorems and Fatou's Lemma.
Fubini and Tonelli theorems.
Definition of Lp and lp spaces and concrete examples of L2 Hilbert spaces
2. Fourier Series:
Definitions and properties.
Sobolev spaces H^s of periodic functions on torus for s real.
Poisson summation/integral formula for the disk and the Dirichlet problem.
3. Bounded linear operators on Hilbert space:
Orthogonal projection.
Dual space of Hilbert space and representation theorems.
Weak convergence in Hilbert space.
Banach-Alaoglu Theorem
4. Spectrum of bounded linear operators:
Diagonalization of matrices.
Spectral theorem for compact, self-adjoint operators.
Compact operators.
Fredholm Alternative Theorem.
Functions of operators.
5. Calculus on Banach Space:
Bochner integrals.
Derivatives of maps on Banach spaces.
The calculus of variations.
Textbooks:
Analysis by Elliott H. Lieb and Michael Loss
and
Applied Analysis by John K. Hunter and Bruno Nachtergaele
Pdf Files of Applied Analysis
Homework:
Homework will be assigned online each
Friday, due next Friday by 09 AM (there will be no homework during the midterm exam week ).
Homework 1 (due Friday, January 15th by 09:00AM):
Exercises 1.1, 1.2, 1.3, 1.4, 1.5 (page 37 of Lieb and Loss).
Solutions (pdf file).
Homework 2 (due Friday, January 29 by 09:00AM):
Exercises 1.9, 1.10, 1.12, 1.13, 1.17, 1.18 (pages 37-39 of Lieb and Loss) plus problems 6.1, 6.3, 6.5, 6.8, 6.12 from Hunter and Nachtergaele
(pages 144-145) plus prove that the convolution of two continuous functions on the unit circle is continuous.
Solutions (pdf file).
Homework 3 (due Friday, February 5 by 09:00AM):
Exercises 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7 (pages 182-184) from the H-N textbook.
Solutions (pdf file).
Homework 4 (due Friday, February 12 by 09:00AM):
Exercises 7.9, 7.10, 7.14, 7.15, 7.17, 7.18 from the H-N textbook.
Solutions (pdf file).
Homework 5 (due Wed, February 24 by 09:00AM):
Exercises 8.2, 8.3, 8.4, 8.6, 8.7, 8.8, 8.9, 8.11, 8.12 from the H-N textbook.
Solutions (pdf file).
Homework 6 (due Friday, March 4 by 09:00AM):
Exercises 8.1, 8.10, 8.13, 8.14, 8.15, 8.16, 8.17, 8.18, 8.19 from the H-N textbook.
Solutions (pdf file).
Homework 7 (due Monday, March 14 by 09:00AM):
Exercises 9.1, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.18 from the H-N textbook.
Lectures:
- Lecture 01 (January 04): Introduction. Basic Notions of Measure Theory. (Sections 1.1 and 1.2 of Lieb and
Loss).
- Lecture 02 (January 06): Basic Notions of Measure Theory. Monotone Class Theorem.
(Sections 1.2 and 1.3 of Lieb and Loss).
- Lecture 03 (January 08): Monotone Class Theorem. Uniqueness of Measures (Sections 1.3 and 1.4
of Lieb and Loss).
- Lecture 04 (January 11): Definition of Measurable Functions and Integrals (Section 1.5 of Lieb and Loss).
- Lecture 05 (January 13): Monotone Convergence. Fatou's Lemma. Dominated Convergence
(Sections 1.6-1.8 of Lieb and Loss).
- Lecture 06 (January 15): Product Measure. Fubini Theorem (Sections 1.10-1.12 of Lieb and Loss).
- Lecture 07 (January 20): L^P Spaces (Section 2.1 of Lieb and Loss).
Approximate Identity. Approximation of L^2 functions by trigonometric polynomials (Section 7.1 of Hunter-Nachtergaele).
- Lecture 08 (January 22): Unconditional and Absolute Convergence of Unordered Sums
(Section 6.3 of Hunter-Nachtergaele).
- Lecture 09 (January 25): Bessel Inequality. Orthonormal Bases(section 6.3 og H-N).
- Lecture 10 (January 27): Parseval's Identity. Existence of an Orthonormal Basis (section 6.3 og H-N).
- Lecture 11 (January 29): The Fourier Basis. Convolution. (Section 7.1 of H-N).
- Lecture 12 (February 1): The Fourier Basis. Convolution. (Section 7.1 of H-N).
- Lecture 13 (February 3): Fourier Series of Differentiable Functions. Sobolev Spaces. Sobolev Embedding.
(Section 7.2 of H-N).
- Lecture 14 (February 5): Weak Derivative. The Heat Equation. (Sections 7.2 and 7.3 of H-N).
- Lecture 15 (February 8): More Applications of Fourier Series. Weyl Ergodic Theorem (Section 7.5 of H-N).
- Lecture 16 (February 10): Wavelets (Section 7.6 of H-N).
- Midterm (February 12).
- Lecture 17 (February 17): Orthogonal Projections (Section 8.1).
- Lecture 18 (February 19): The dual of a Hilbert space (Section 8.2).
- Lecture 19 (February 22): The adjoint of an operator (Section 8.3).
- Lecture 20 (February 24): The adjoint of an operator (Section 8.3).
- Lecture 21 (February 26): Self-adjoint and unitary operators (Section 8.4).
- Lecture 22 (February 29): The mean ergodic theorem (Section 8.5).
- Lecture 23 (March 2): Weak convergence ina Hilbert space (Section 8.6).
- Lecture 24 (March 4): The spectrum of bounded linear operators (Sections 9.1 and 9.2).
- Lecture 25 (March 7): The spectrum of bounded linear operators (Sections 9.1 and 9.2).
- Lecture 26 (March 9): The spectral theorem for compact, self-adjoint operators (Section 9.3).
- Lecture 27 (March 11): Compact operators (Section 9.4).
- Lecture 28 (March 14): Differential Calculus and Variational Methods (Chapter 13).