# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 200B: Problem-Solving in Analysis

Approved: 2010-10-01,
Suggested Textbook: (actual textbook varies by instructor; check your instructor)

Prerequisites:
Courses MAT 201ABC or consent of instructor.
Course Description:
Sequence Description: Problem-solving in graduate an alysis: continuous functions, metric spaces, Banach and Hilbert spaces, bounded linear operators, the spectral theorem, distributions, Fourier series and transforms, Lp spaces, Sobolev spaces.
Suggested Schedule:

Summary of Sequence Content –

This sequence is intended as a workshop on solving problems in Analysis. The topics covered include the following:

1. Continuous functions. Convergence of functions. Spaces of continuous functions. Stone-Weierstrass theorem. Arzela-Ascoli theorem.
2. Metric spaces. Completeness. Contraction Mapping Theorem and its applications.
3. Normed spaces. Banach spaces. Bounded linear operators. Different types of operator convergence. Compact operators. Dual spaces. Finite-dimensional Banach spaces.
4. Hilbert spaces. Orthogonality. Orthonormal bases. Parseval's identity. The dual of a Hilbert space (Riesz representation theory).
5. Bounded linear operators on a Hilbert space. Orthogonal projections. The adjoint of an operator. Self-adjoint and unitary operators.
6. Spectrum and Resolvent. The spectral theorem for compact self-adjoint operators. Hilbert-Schmidt operators.
7. The Fourier basis. Fourier series. Convolution. Fourier series of differentiable functions.
8. Distributions. The Schwartz space. Tempered distributions. Operations on distributions.
9. The Fourier transform. The Fourier transform of test functions and of distributions. The Fourier transform on L1 and L2 spaces. Riemann-Lebesgue Lemma. Plancherel Theorem.
10. L^p spaces. The dual space of L^p. Basic inequalities: Jensen inequality; Holder inequality; Minkowski inequality; Chebyshev inequality; Young inequality.
11. Sobolev spaces. Weak derivatives. Sobolev Embedding theorem. Rellich-Kondrachov theorem. Poincare inequality. Laplace's equation.
Assessment:
200A (Spring) and 200B (the following Fall) are offered as a sequence with deferred grading in Spring. The final grade for the sequence will be recorded at the end of Fall. 200A will have 1 unit; 200B will have 2 units. May be repeated once.