## COURSE DESCRIPTION

Real numbers, completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, the fundamental theorem of calculus. Vector spaces, dimension, linear maps, inner products and norms. This course in intended to assist CAAM graduate students in preparing for the qualifying exams. See the syllabus.

**Text Book:** *Undergraduate Analysis* by Lang

**Homeworks:**
Homework assignments will be given weekly. They will be posted on Wednesdays and will be due the following Tuesday. Two assignments will be pledged and will each count as 20% of the grade. The other homework assignments will count for 50% of the grade. Classroom participation will count as 10% of the grade. Homeworks should be handed in on time. One homework will be accepted up to a week late without consequence. In that time, you may not look at the posted solutions. Subsequent late submissions will not be accepted, unless by prior permission of the instructor or by a truly exceptional circumstance.

**Classroom participation:**
Each day of class, one or two students will present a theorem from that day’s content. The classroom participation grade will be based on the sincerity of your preparation for these presentations. Please follow my comments and suggested structure for the presentations.

**Outside resources:**
You are not allowed to use the Problems and Solutions book accompanying Lang’s Undergraduate Analysis text for any of the homeworks.

**Disabilities:**
Any student with a disability needing academic accommodations is requested to speak with me as soon as possible. All discussions will remain confidential. Students should also contact Disability Support Services in the Ley Student center.

# SCHEDULE

Event | Date | Related Documents |
---|---|---|

HW 1 | Sep 2 in class | Problems. Solutions |

HW 2 | Sep 9 in class | Problems. Comments on presentations. Example of presentation notes. Solutions |

HW 3 | Sep 16 in class | Problems. Solutions |

HW 4 | Sep 23 in class | Problems Solutions |

Optional HW 5 | Sep 30 in class | Problems |

Pledged HW 6 | Oct 7 in class | Problems |

HW 7 | Oct 21 in class | Problems Solutions |

HW 8 | Oct 28 in class | Problems Solutions |

HW 9 | Nov 4 in class | Problems. Solutions |

HW 10 | Nov 11 in class | Problems Solutions |

HW 11 | Nov 18 in class | Problems Solutions |

HW 12 | Nov 25 in class | Problems Solutions |

Pledged HW 13 | Dec 4 in class | Problems |

# Lecture Notes

Day | Topics | Reference (Lang) | Class notes |
---|---|---|---|

Aug 26 | Real numbers. | I.1-I.4, II.1 | Summary |

Aug 28 | Cauchy sequences, Bolzano-Weierstrass, liminf and limsup | II.1, II.2 | Summary. |

Sep 2 | Limits, continuity | II.2, II.4 | Summary |

Sep 4 | Squeeze theorem, limits with infinity | II.2, II.3, III.1 | Summary |

Sep 9 | Differentiability, Mean value theorem | III.2, III.3 | Summary |

Sep 11 | Convex funcions, inverse function theorem | III.2, III.3 | Summary |

Sep 16 | Riemann Integration | V.1-V.2 | Summary |

Sep 18 | Riemann Integration, Taylor Series | V.2-V.3 | Summary |

Sep 23 | Taylor Series, Normed vector spaces | V.3, VI.1-VI.3 | Summary |

Sep 25 | Normed vector spaces | VI.2-VI.3 | Summary |

Sep 30 | |||

Oct 2 | |||

Oct 7 | Banach spaces | VI.4 | Summary |

Oct 9 | Open and closed sets | VI.5 | Summary |

Oct 16 | Dimensionality of spaces | Notes by Symes | Summary |

Oct 21 | Limits in normed vector spaces and function spaces | VII.1, VII.2, VII.3 | Summary |

Oct 23 | Equivalence relations | VII.4 | Summary |

Oct 28 | Completion of spaces | VII.4 | Summary |

Oct 30 | Sequential compactness, Compactness by open covers | VIII.1, VIII.2, VIII.4 | Summary |

Nov 4 | Series | IX.1, IX.2, IX.3 | Summary |

Nov 6 | Absolute and uniform convergence | IX.4, IX.5 | Summary |

Nov 11 | Power series | IX.6-IX.7 | Summary |

Nov 13 | Extension of linear functions | X.1, X.2, X.3 | Summary |

Nov 18 | Integral via step functions | X.1-X.3 | Summary |

Nov 20 | Measure and content | X.4 appendix | Summary |

Nov 25 | Almost everywhere convergence | X.4 appendix | Summary |

Dec 2 | Relation of derivatives and integrals | X.5-X.7 | Summary |

Dec 4 | Lebesgue Integral | X.4 appendix | Summary |