The lecture topics have accompanying lecture summaries.

LEC # | TOPICS | KEY DATES |
---|---|---|

1 | Sets, ordered sets, countable sets | |

2 | Fields, ordered fields, least upper bounds, the real numbers | |

3 | The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz | Problem set 1 due |

4 | Metric spaces, ball neighborhoods, open subsets | |

5 | Open subsets, limit points, closed subsets, dense subsets | Problem set 2 due |

6 | Compact subsets of metric spaces | |

7 | Limit points and compactness; compactness of closed bounded subsets in Euclidean space | Problem set 3 due |

Midterm | ||

8 | Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy's theorem | |

9 | Subsequential limits, lim sup and lim inf, series | Problem set 4 due |

10 | Absolute convergence, product of series | |

11 | Power series, convergence radius; the exponential function, sine and cosine | Problem set 5 due |

12 | Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps | |

13 | Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity | Problem set 6 due |

14 | Derivatives, the chain rule; Rolle's theorem, Mean Value Theorem | |

15 | Derivative of inverse functions; higher derivatives, Taylor's theorem | Problem set 7 due |

Midterm | ||

16 | Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series | |

17 | Uniform convergence of derivatives | |

18 | Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem | Problem set 8 due |

19 | End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) | |

20 | Riemann-Stjeltjes integral: definition, basic properties | |

21 | Riemann integrability of products; change of variables | Problem set 9 due |

22 | Fundamental theorem of calculus; back to power series: continuity, differentiability | |

23 | Review of exponential, log, sine, cosine; e+^{it}= cos(t) isin(t) | Problem set 10 due |

24 | Review of series, Fourier series | |

Final Exam |