### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Prerequisites

Analysis I (18.100); Linear Algebra (18.06), Linear Algebra (18.700), or Algebra I (18.701)

### Topics

The course will be in three approximately equal parts (so about 4 weeks each).

- Normed spaces and a brief treatment of integration
- Norms, bounded linear operators, completeness
- Step functions, covering lemma, Lebesgue integrable functions
- Fatou’s lemma, dominated convergence, L1

- Hilbert space
- Cauchy’s inequality, Bessel’s inequality, orthonormal bases
- Convex sets, minimization, Riesz’ theorem, adjoints
- Compact sets, weak convergence, Baire’s theorem, uniform boundeness

- Operators on Hilbert space
- Finite rank and compact operators
- Spectral theorem for compact self-adjoint operators
- Fourier series, periodic functions
- Dirichlet problem on the interval, completeness of eigenfunctions

### Grading

Grades will be computed by two methods — the cumulative and the hope-springs-eternal method with the actual grade the greater of the two.

- First method: Homework 30, Tests 30, Final 40
- Second method is based purely on the final