## COURSE DESCRIPTION

The objectives of this course are: (1) to provide you with a firm understanding of the basic ideas of mathematical analysis; (2) to improve your proof writing skills up to a professional level; (3) to improve problem solving skills needed for the CAAM analysis qualifier exam; and (4) to improve your mathematical communication skills. The specific analysis topics covered include 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, derivatives in R^d, Inverse Function Theorem, Implicit Function Theorem, Multiple Integration, Change of Variable Theorem. See the syllabus.

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

**Homeworks:**
You will have about three homework problems a week. They will be due on Tuesdays. Your proofs should be your absolute best quality work. They will each be graded out of one point, taking into account the correctness and the exposition. After you receive your graded solution, you may resubmit it once (at the next homework due date). The grade of the second submission will be final. Two homework assignments will be pledged and will serve as simulated qualifier exams. Here are some examples of the quality of proofs expected. Two homework assignments will be pledged and will serve as simulated qualifier exams.

**Class Time:**
Class will primarily consist of a discussion and Q&A on the topics assigned that week. It will also include some problem solving and presentation of solutions to homework problems. You are expected to read and think deeply about the day’s topics before the beginning of class. During the discussion, I will choose students randomly to present and answer questions on that day’s topics. See these comments for the sorts of things you should think about.

**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 | Aug 29 in class | Problems. |

HW 2 | Sep 5 in class | Problems. |

HW 3 | Sep 12 in class | Problems. |

HW 4 | Sep 26 in class | Problems. |

HW 5 | Oct 3 in class | Problems. |

Pledged HW 6 | Oct 19 in class | Exam 2017 with solutions. Practice Exam 2016. Practice Exam 2015 with solutions. Practice Exam 2014. |

HW 7 | Oct 24 in class | Problems. |

HW 8 | Oct 31 in class | Problems |

HW 9 | Nov 7 in class | Problems. |

HW 10 | Nov 14 in class | Problems. |

Pledged HW 11 | Nov 30 in class | Exam 2017. Final from 2016. Final from 2014. Final from 2015 |

# Lecture Notes

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

Aug 22,24 | Real numbers. Cauchy sequences, Bolzano-Weierstrass, liminf and limsup. Limits, continuity. | I.1-I.4, II.1-4 | Notes |

Sep 5,7 | Differentiation, Mean value theorem, Taylor Series | III.2-III.3,V.3 | Notes |

Sep 12,14 | Riemann Integration | V.1-V.2 | Notes |

Sep 19,21 | Normed Vector Spaces | VI.1-VI.3 | Notes |

Sep 26,28 | Normed Vector Spaces, Dimensionality of Spaces | VI.1-VI.5, | Notes on dimensions. Notes |

Oct 3,5 | Dimensionality of spaces, Limits in normed vector spaces and function spaces | VII.1, VII.2,VII.3 | Notes |

Oct 12 | Completion of Spaces | VII.4 | Notes |

Oct 17,19 | Open and closed Sets, Compactness | VI.5, VIII.1, VIII.2, VIII.4 | Notes |

Oct 24,25 | Series and Power Series | IX.1--IX.7 | Notes |

Oct 31, Nov 2 | Integral in One Variable | X.1-X.7 | Notes |

Nov 7,9 | Derivatives in Vector Spaces | XV.1-XV.2, XVII.1-XVII.3. | Notes |

Nov 14,16 | Inverse Function Theorem | XVIII | Notes |

Nov 21 | Implicit Function Theorem | XVIII | Notes |

Nov 28, Nov 30 | Measure zero and content zero | X.4 Appendix | Notes |