Introduction to Functional Analysis

Syllabus

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).

1. 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
2. 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
3. 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