Fall 2019 MAT 551: Analysis III, Functional analysis | ||

Schedule | MW 2:30-3:50 Physics P123 | |

Instructor | Robert Hough | |

Office hours | W 6-7 in MLC, F 9am-11pm in Math Tower 4-118. | |

Description | Selections from the following topics. Partial differential equations in higher dimensions; Sobolev spaces, calculus of variations, characteristics, Cauchy problem, energy estimates, maximum principles, Harmonic analysis; singular integrals, Hausdorff measure, harmonic measure, Hardy spaces, Functional analysis; spectral theory, distributions, Banach algebras. | |

Textbook | Peter Lax. Functional Analysis. John Wiley and Sons (2002). | |

Supplementary Textbook | Michael Reed and Barry Simon. Functional Analysis. John Wiley and Sons (1980). | |

Homework | Weekly problem sets will be assigned, and collected in class on Monday. | |

Grading | The course grade is based upon the written homework. |

** Syllabus/schedule (subject to change)**

Mon 8/26 | 1. | Linear spaces | Lax Chapters 1-4 |

Wed 8/28 | 2. | Hahn Banach | HW due 9/2: p. 12 #4,5, p. 32 #1,2 |

Mon 9/2 | 3. | Normed linear spaces | Lax Chapter 5 |

Wed 9/4 | 4. | Normed linear spaces | HW due 9/9: p.39 #3, p. 44 #4, p. 48 #8 |

Mon 9/9 | 5. | Hilbert Space | Lax Chapters 6-7 |

Wed 9/11 | 6. | Applications of Hilbert Space | HW due 9/16: p.53 #1, p.54 #3, p.62 #10, p.65 #2 |

Mon 9/16 | 7. | Duality | Lax Chapters 8-9 |

Wed 9/18 | 8. | Applications of duality | HW due 9/23: p.76 #1,2, p.77 #3,4, p.84 #6 |

Mon 9/23 | 9. | Weak convergence | Lax Chapters 10-12 |

Wed 9/25 | 10. | Weak and weak* topologies | HW due 9/30: p.101 #2, p.104 #4, p.106 #6 |

Mon 9/30 | 11. | Convexity | Lax Chapters 13-14 |

Wed 10/2 | 12. | Extreme points | HW due 10/7: p.122 #1,2, p.129 #7, p.144 #4, p.150 #5, p.152 #6 |

Mon 10/7 | 13. | Bounded linear maps | Lax Chapters 15-16 |

Wed 10/9 | 14. | Examples of bounded linear maps | HW due 10/16: p.164 #2, p. 165 #5, p.166 #6, p.171 #12, p.184 #3, p.185 #4 |

Mon 10/14 | No class - Fall Break | ||

Wed 10/16 | 15. | Banach algebras | Lax Chapter 17, HW due 10/21: p.201 #1-5 |

Mon 10/21 | 16. | Spectrum | Lax Chapters 19-20 |

Wed 10/23 | 17. | Examples of operators | HW due 10/28: p.214 #2, p.219 #3, p.229 #1, p.230 #2,3, p.231 #4,6, p.232 #7 |

Mon 10/28 | 18. | Compact operators | Lax Chapters 21-23 |

Wed 10/30 | 19. | Examples of compact operators | HW due 11/4: p.234 #2, p.240 #4, p.241 #5, p.242 #6, p.243 #8 |

Mon 11/4 | 20. | Fredholm Theory | Lax Chapters 24-26 |

Wed 11/6 | 21. | Invariant subspaces | HW due 11/11: p.265 #2, p.271 #6 |

Mon 11/11 | 22. | Index theory, compact symmetric operators | Lax Chapters 27-29 |

Wed 11/13 | 23. | Examples | HW due 11/18: p.312 #1, p.313 #2,3, p.318 #4, 5, p.321 #7, 8, p.327 #2 |

Mon 11/18 | 24. | Trace class, trace formula | Lax Chapter 30 |

Wed 11/20 | 25. | Examples | HW due 11/25: p.330 #2, p.333 #4, p.342 #5, p.347 #8 |

Mon 11/25 | 26. | Spectral theory | Lax Chapter 31 |

Wed 11/27 | No class - Thanksgiving | ||

Mon 12/2 | 27. | Self-adjoint operators | Lax Chapters 32-33 |

Wed 12/4 | 28. | Examples | HW due 12/9: p.372 10-12, p.390 #1, p.405 #6,7 |

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