Date | Topics |
---|---|

R 9/2 | Introduction. Weierstrass existence theorem. |

T 9/7 | Finite dimensional spaces. Riesz lemma. Uniform convexity. |

R 9/9 | Hilbert spaces. Projection. Riesz representation. |

T 9/14 | Sobolev spaces. Strong derivatives. |

R 9/16 | Mollifiers. Du Bois-Reymond lemma. Weak derivatives. |

T 9/21 | Meyers-Serrin theorem. Sobolev lemma. Regularity. |

R 9/23 | Neumann problems. Rellich lemma. Complex scalars. |

T 9/28 | Orthonormal bases. |

R 9/30 | Fourier bases. |

T 10/5 | Sturm-Liouville problems. Resolvent formalism. |

R 10/7 | Hilbert-Schmidt theory. |

F 10/15 | Gelfand triples. Singular Sturm-Liouville problems. |

T 10/19 | Functional calculus. Evolution equations. |

R 10/21 | Seminorms. Local spaces. Tempered distributions. |

T 10/26 | Hahn-Banach theorem. |

R 10/28 | Milman-Pettis theorem. |

T 11/2 | Weak convergence. Uniform boundedness principle. |

R 11/4 | Banach-Alaoglu theorem. |

T 11/9 | Tychonov's theorem. Consequences of the open mapping theorem. |

R 11/11 | Open mapping theorem. Bounded below property. Range closedness. |

T 11/16 | Closed range theorem. |

R 11/18 | Inf-sup conditions. Saddle point problems. |

T 11/23 | Perturbations of invertible operators. Fredholm alternative. |

R 11/25 | Compact operators. |

T 11/30 | Riesz-Schauder theory. Fredholm operators. Regularizers. |

R 12/2 | Perturbations of Fredholm operators. Bounded self-adjoint operators. |

- Barbara MacCluer,
*Elementary Functional Analysis*. Springer. - Martin Schechter,
*Principles of Functional Analysis*. AMS. - Peter Lax,
*Functional Analysis*. Wiley.

- Hahn-Banach theorem, duality
- Weak and weak-star convergence
- Bounded linear operators in Banach spaces
- Compact operators, Fredholm theory
- Spectral theory of bounded self-adjoint operators
- Rudiments of locally convex spaces and unbounded operators
- Semigroups (if time permits)

**Instructor:** Dr. Gantumur Tsogtgerel

**Prerequisite:** MATH 455 (Honours Analysis 4) or equivalent

**Calendar description:**
Banach and Hilbert spaces,
theorems of Hahn-Banach and Banach-Steinhaus,
open mapping theorem,
closed graph theorem,
Fredholm theory,
spectral theorem for compact self-adjoint operators,
spectral theorem for bounded self-adjoint operators.

**Grading:** Homework assignments 40% + Midterm Exam 20% + Course project 40%.

**Homework:** Assigned and graded roughly every other week, through MyCourses.

**Midterm:** Take-home type, in late Octorber or early November.

**Course project:** The course project consists of the student reading a paper or monograph on an
advanced topic, typing up notes, and perhaps giving a lecture.