Class

lecture | tutorial | |

Time | Tue, Thu | Wed |

13:30 - 14:50 | 20:00 - 21:00 | |

Venue | 2463 | 2404 |

Instructor | YAN Min | |

Office | 3487 | |

Phone | 23587442 |

Lecture Note

My own lecture note will be posted on-line. No textbook from the market will be used.

Schedule

Week 1

L1: Norm, Limit of Sequence in **R**^{n}

L2: Multivariable Map, Repeated Limit, Continuity

Week 2

L1: Compact Subset, Properties of Continuous Map, Closed Subset, Equivalence of Norm

L2: Open Subset, Topology of **R**^{n}, Set Theoretical Interpretation of Continuity

Week 3

L1: Linear Transform, Bilinear Map, Norm of Linear Transform and Bilinear Map

L2: Multilinear Algebra, Determinant, Orientation

Week 4

L1: Linear Approximation, Chain Rule, Mean Value Theorem

L2: Inverse and Implicit Differentiation

Week 5

L1: Submanifold

L2: High Order Approximation, Order of Differentiation

T: Maximum and Minimum, Lagrange Multiplier

Week 6

L1: Length, Lebesgue Measure on **R**, Cantor Set

L2: Caratheodory Theorem

Week 7

L1: Abstract Outer Measure, σ-Algebra, Abstract Measure

L2: Lebesgue Integral

T: Set of Measure Zero, Complete Measure

Week 8

L1: Measurable Function, Criterion for Riemann Integrability

L2: Unbounded Integral, Simple Function Approximation

Week 9

L1: Convergence Theorem

L2: Extension Theorem

T: Approximation Theorem: Egorov, Lusin, etc.

Week 10

L1: Lebesgue-Stieltjes Measure, Regular Measure on **R**

L2: Product Measure, Fubini Theorem

Week 11

L1: Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem

L2: Lebesgue Differentiation Theorem

T: Lesbegue Measure on **R**, Translation Invariance, Non-Lesbegue Measurable Set, Cantor Function

Week 12

L1: Fundamental Theorem of Lebesgue Integral on **R**

L2: Rademacher Theorem, Change of Variable on **R**^{n}

Homework

No hard copies of the homework will be distributed. Please check my web page for the latest assignments. Usually you have one week to do the homework. The answer will be put on my web page after the homework is collected. *Absolutely no late homework will be accepted*.

Exam and Grade

There will be one midterm and one final exam, accounting for 30% and 50% respectively of the final grade. The homework is counted as 20% of the final grade.

Note

The syllabus is subject to change as circumstances arise.

As my usual practice, I try to make the homework as light as possible. However, doing homework alone is definitely not sufficient for really understanding the subject. You should do as many additional problems as possible.