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The syllabus will be updated as the course progresses. Unless otherwise indicated, section numbers and exercises refer to the text *A Course in Modern Analysis and its Applications* by Graeme L. Cohen.

Date |
Text Sections |
Material
| Recommended Exercises ^{*} |

Sep 9 | 2.1, 2.2 | Metric spaces: definitions and examples | Section 2.4: (1)–(5) |

Sep 14 | 2.2, 1.10, 2.5 | More examples, convergence in a metric space | Section 2.4: (6)–(8), (10)–(12) |

Sep 16 | 1.10, 2.5, 2.6 | Convergence in a metric space, sequences of functions, complete metric spaces | Section 2.9: (1), (2), (5)–(8) |

Sep 21 | 2.6, 2.7, 2.8 | Complete metric spaces, metric subspaces | Section 2.9: (3), (4), (9)–(12), (14) |

Sep 23 | 3.1, 3.2 | Mappings between metric spaces, The Fixed Point Theorem | Section 3.5: (3), (8), (10) |

Sep 28 | 3.3, 4.1 | Applications of The Fixed Point Theorem, compact sets | Section 3.5: (2)–(4); Section 4.5: (1)–(5) |

Sep 30 | 4.2, 4.3 | Arzelà-Ascoli Theorem | Section 4.5: (7)–(9) |

Oct 4 | 4.3, 5.1 | Approximation theory, topological spaces | Section 5.7: (1), (2) |

Oct 7 | 5.1, 5.2 | The metric topology, closed sets | Section 5.7: (4a), (5) |

Oct 12 | 5.2 | Closed sets, separable metric spaces | Exercises in lecture notes only. |

Oct 14 | 5.3, 5.5 | Connectedness, compactness | Section 5.7: (7), (8) |

Oct 19 | N/A | Midterm exam (covers material up to including lecture of October 12) |
N/A |

Oct 21 | 5.3 | Compactness | Section 5.7: (15) |

Nov 2 | 5.4, 5.5 | Continuity, path-connectedness | Section 5.7: (9) |

Nov 4 | 5.5 | Path-connectedness | Exercises in lecture notes only. |

Nov 9 | 6.1 | Completions, normed vector spaces | Section 6.4: (1b), (2), (4), (12) |

Nov 11 | 6.2, 6.5 | Convergence in normed spaces, finite-dimensional normed spaces | Section 6.4: (3), (5)–(8), (11); Section 6.10: (2) |

Nov 16 | 6.5, 6.6 | Student course evaluations. Finite-dimensional normed spaces, approximation theory. |
Section 6.10: (3), (4) |

Nov 18 | 6.6 | Weierstrass approximation. | Section 6.10: (5), (12)–(15) |

Nov 23 | 7.1, 7.2 | Bounded linear mappings | Section 7.5: (2), (3), (9), (10), (13) |

Nov 25 | 7.2, 7.3 | Bounded linear mappings, linear functionals | Section 7.5: (6), (7), (12), (14) |

Nov 30 | 8.1, 8.2 | Hahn-Banach Theorem, inner product spaces | Section 8.6: (1)–(10) |

Dec 2 | 9.1, 9.2 | Hilbert space and Fourier series | Section 9.8: (1), (3), (4) |

Dec 7 | N/A | Review | N/A |