Fall 2018 MATH 5200
Undergraduate Analysis 初等分析

Syllabus 課程大綱

宣導事項

教學進度

  • 9/12/2018(星期三): Introduction/Motivation to Lebesgue Theory.
  • 9/14/2018(星期五): General introduction to measure theory, existence of non-measurable set.
  • 9/19/2018(星期三): Sigma algebras.
  • 9/21/2018(星期五): Set functions and outer measures, basic properties.
  • 9/26/2018(星期三): Basic Properties of outer measure.
  • 9/28/2018(星期五): Construction of sigma algebra of measurable subsets.
  • 10/3/2018(星期三): Borel sets, sets of measure 0.
  • 10/5/2018(星期五): Measurable functions.
  • 10/10/2018(星期三): 台灣國慶日放假。
  • 10/12/2018(星期五): Approximations by simple functions, integrations, basic properties.
  • 10/17/2018(星期三): Measure defined by integrals.
  • 10/19/2018(星期五): Monotone convergence theorem.
  • 10/24/2018(星期三): Fatou's lemma, dominated convergence theorem.
  • 10/26/2018(星期五): Comparisons with Riemann integrals.
  • 10/31/2018(星期三): Introduction to L^p spaces.
  • 11/2/2018(星期五): Completeness of L^p spaces.
  • 11/7/2018(星期三): Density of step functions in L^p.
  • 11/9/2018(星期五): Midterm 1.
  • 11/14/2018(星期三): Intdoduction to Hilbert spaces.
  • 11/16/2018(星期五): Basic properties of Hilbert spaces.
  • 11/21/2018(星期三): More properties of Hilbert spaces.
  • 11/23/2018(星期五): Separability and orthonormality.
  • 11/28/2018(星期三): Fourier series.
  • 11/30/2018(星期五): Fourier series and convergences.
  • 12/5/2018(星期三): Introduction to Fourier transforms.
  • 12/7/2018(星期五): Convolutions.
  • 12/12/2018(星期三): Fourier transforms and basic properties.
  • 12/14/2018(星期五): Fourier transoforms for more general functions.
  • 12/19/2018(星期三): Heat equation.
  • 12/21/2018(星期五): Heat equation and Dirichlet problem.
  • 12/26/2018(星期三): Signal analysis.
  • 12/28/2018(星期五): Signal analysis and sampling theorem.
  • 1/2/2019(星期三): Dispersion and Heisenberg's inequality.
  • 1/4/2019(星期五): Quamtum mechanics and uncertainty principle.

    作業

  • Homework 1 (截止日 9/28/2018 in Class):
    Download Here Solutions
  • Homework 2 (截止日 10/12/2018 in Class):
    Download Here Solutions
  • Homework 3 (截止日 11/9/2018 in Class):
    Download Here Solutions
  • Homework 4 (截止日 11/30/2018 in Class):
    Download Here
  • Homework 5 (截止日 12/21/2018 in Class):
    Folland Section 7.1: 2, 3, 4, 5, 8.
    Folland Section 7.2: 2~7.

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