Analysis I, Fall 2020

Course Information

• Ceiba
• Lectures: Tuesday, 10:20 ~ 12:10 at Astro-Math 102
  Thursday 10:20 ~ 12:10 at Astro-Math 102
• Problem Session: Tuesday, 9:10 ~ 10:00 at Astro-Math 102 (林自立)
  and Astro-Math 101 (李宸寬)
• Office hour: Tuesday 14:00 ~ 15:00, at Astro-Math 458
• Grading scheme:
  • Homework 35%
    You have two jokers: the lowest two grades will be discarded.
  • Quiz 15% (on Sept 22, Oct 13, Dec 1, Dec 22)
  • Midterm 25% (on Nov 10)
  • Final Exam 25% (on Jan 5)
• Course prerequisite:
  • freshman calculus (for math major)
  • linear algebra (for math major)
• Textbook: Charles Chapman Pugh, Real mathematical analysis. Second Edition. MR
  You can download the PDF files with an NTU IP address. link
• Teaching Assistants: Please find on the Ceiba website for the information of their office hours.
  • 林自立
  • 李宸寬

Lecture summaries

• (Week 1) metric space, continuity, open and closed set. Reference: §2.1, §2.2, §2.3
• (Week 2) completeness, compactness, connectedness. Reference: §2.4, §2.5
• (Week 3) covering and compactness, Cantor set. Reference: §2.7, §2.8
• (Week 4) Cantor set, uniform convergence, integration and differentiation of sequence of functions, power series. Reference: §2.8, §2.8, §4.1, §4.2
• (Week 5) Arzela-Ascoli theorem, Weierstrass approximation theorem. Reference: §4.3, §4.4
• (Week 6) Stone-Weierstrass theorem, Picard's theorem for ODE, nowhere differentiable function. Reference: §4.4, §4.5, §4.7
• (Week 7) Baire category theorem, space of continuous functions over non-compact spaces, σ-compact and hemi-compact, Urysohn lemma and Tietze extension, continuous dependence of ODEs on initial conditions. Reference: §4.7, §4.8
• (Week 8) completion, paracompactness. derivatives and higher order derivatives in higher dimensions. Reference: §2.10, §5.1, §5.2, §5.3
• (Week 9) inverse and implicit function theorem, constant rank theorem. Reference: §5.4, §5.5
• (Week 10) Fall Break.
• (Week 11) exterior algebra, differential forms, Stokes theorem. Reference: §5.8, §5.9
• (Week 12) more on the Stokes theorem, Poincaré lemma, Brouwer fixed point theorem, Perron--Frobenius theorem. Reference: §5.9, §5.10
• (Week 13) Lebesgue measure, abstract outer measure, regularity. Reference: §6.1, §6.2, §6.4
• (Week 14) product and slice, Lebesgue integral, monotone convergence theorem. Reference: §6.5, §6.6
• (Week 15) dominated convergence theorem, Vitali covering lemma, density. Reference: §6.7, §6.8
• (Week 16) Lebesgue's fundamental theorem of calculus, absolute continuity. Reference: §6.9, §6.10
• (Week 17) differentiability of monotone function, Littlewood's Three Principles. Reference: §6.10, Appendix F

Homework

• Homework 00 (no due). Ch.2 exercise #5, #6, #12, #13, #116, #119.
• Homework 01 (due Sept 29). Ch.2 exercise #39, #44, #46, #57, #71.
• Homework 02 (due Oct 6). Ch.2 exercise #28, #52, #53, #86, #89, #92.
• Homework 03 (due Oct 13). Ch.2 exercise #54, #109. prelim problems #3, #5. Ch.4 exercise #2.
• Homework 04 (due Oct 20). Ch.4 exercise #7, #8, #10, #11, #17.
• Homework 05 (due Oct 27). Ch.4 exercise #27, #28, #35. prelim problems #34, #60.
• Homework 06 (due Nov 3). Ch.4 exercise #34, #40. prelim problems #12, #19, #58.
• Homework 07 (due Nov 24). Ch.5 exercise #8, #14, #15, #30, #43.
• Homework 08 (due Dec 1). see the attached file on ceiba.
• Homework 09 (due Dec 8). Ch.5 exercise #56, #58, #65, #66, #67, #69.
• Homework 10 (due Dec 15). Ch.6 exercise #5, #10, #11, #12, #13.
• Homework 11 (due Dec 22). Ch.6 exercise #21, #22, #23, #25.
• Homework 12 (due Dec 29). Ch.6 exercise #29, #31, #33, #36, #53, #58.