## 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 30 in class | Problems. |

HW 2 | Sep 6 in class | Problems. |

HW 3 | Sep 13 in class | Problems. |

HW 4 | Sep 27 in class | Problems. |

HW 5 | Oct 4 in class | Problems. |

Pledged HW 6 | Oct 18 in class | Problems. Practice Exam 2015 with solutions. Practice Exam 2014. |

HW 7 | Oct 25 in class | Problems. |

HW 8 | Nov 1 in class | Problems |

HW 9 | Nov 8 in class | Problems. |

HW 10 | Nov 15 in class | Problems. |

Pledged HW 11 | Dec 8 by noon | Problems. Final from 2014. Final from 2015 |

# Lecture Notes

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

Aug 23,25 | Real numbers. Cauchy sequences, Bolzano-Weierstrass, liminf and limsup. Limits, continuity. | I.1-I.4, II.1-4 | Notes |

Aug 30,Sep 1 | Differentiation, Mean value theorem, Taylor Series | III.2-III.3,V.3 | Notes |

Sep 6,8 | Riemann Integration | V.1-V.2 | Notes |

Sep 13,15 | Normed Vector Spaces | VI.1-VI.3 | Notes |

Sep 20,22 | Normed Vector Spaces, Dimensionality of Spaces | VI.1-VI.5, | Notes on dimensions. Notes |

Sep 27,29 | Dimensionality of spaces, Limits in normed vector spaces and function spaces | VII.1, VII.2,VII.3 | Notes |

Oct 4,6 | Completion of Spaces, Open and closed Sets | VII.4, VI.5 | Notes |

Oct 13 | Compactness | VIII.1, VIII.2, VIII.4 | Notes |

Oct 18,20 | Series and Power Series | IX.1--IX.7 | Notes |

Oct 25,27 | Integral in One Variable | X.1-X.7 | Notes |

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

Nov 8,10 | Inverse Function Theorem | XVIII | Notes |

Nov 15,17 | Implicit Function Theorem | XVIII | Notes |

Nov 22 | Measure zero and content zero | X.4 Appendix | Notes |

Nov 29, Dec 1 | Multiple Integrals | XX |