|13:30 - 14:50||16:30 - 17:20|
My own lecture note will be posted on-line. No textbook from the market will be used.
L1: Norm, Limit of Sequence in Rn
L2: Multivariable Map, Repeated Limit, Continuity
L1: Compact Subset, Properties of Continuous Map, Closed Subset, Equivalence of Norm
L2: Open Subset, Topology of Rn, Set Theoretical Interpretation of Continuity
L1: Linear Transform, Bilinear Map, Norm of Linear Transform and Bilinear Map
L2: Multilinear Algebra, Determinant, Orientation
L1: Linear Approximation, Chain Rule, Mean Value Theorem
L2: Inverse and Implicit Differentiation
L2: High Order Approximation, Order of Differentiation
T: Maximum and Minimum, Lagrange Multiplier
L1: Length, Lebesgue Measure on R, Cantor Set
L2: Caratheodory Theorem
L1: Abstract Outer Measure, σ-Algebra, Abstract Measure
L2: Lebesgue Integral
T: Set of Measure Zero, Complete Measure
L1: Measurable Function, Criterion for Riemann Integrability
L2: Unbounded Integral, Simple Function Approximation
L1: Convergence Theorem
L2: Extension Theorem
T: Approximation Theorem: Egorov, Lusin, etc.
L1: Lebesgue-Stieltjes Measure, Regular Measure on R
L2: Product Measure, Fubini Theorem
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
L1: Fundamental Theorem of Lebesgue Integral on R
L2: Rademacher Theorem, Change of Variable on Rn
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.
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.