# 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 200A: Problem-Solving in Analysis

**Approved:**2010-10-01,

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

**Prerequisites:**

Graduate standing in Mathematics or Applied Mathematics, 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:

- Continuous functions. Convergence of functions. Spaces of continuous functions. Stone-Weierstrass theorem. Arzela-Ascoli theorem.
- Metric spaces. Completeness. Contraction Mapping Theorem and its applications.
- Normed spaces. Banach spaces. Bounded linear operators. Different types of operator convergence. Compact operators. Dual spaces. Finite-dimensional Banach spaces.
- Hilbert spaces. Orthogonality. Orthonormal bases. Parseval's identity. The dual of a Hilbert space (Riesz representation theory).
- Bounded linear operators on a Hilbert space. Orthogonal projections. The adjoint of an operator. Self-adjoint and unitary operators.
- Spectrum and Resolvent. The spectral theorem for compact self-adjoint operators. Hilbert-Schmidt operators.
- The Fourier basis. Fourier series. Convolution. Fourier series of differentiable functions.
- Distributions. The Schwartz space. Tempered distributions. Operations on distributions.
- 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.
- L^p spaces. The dual space of L^p. Basic inequalities: Jensen inequality; Holder inequality; Minkowski inequality; Chebyshev inequality; Young inequality.
- 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.