A N A L Y S I S    I ,    F A L L   2 0 2 0

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 theorm 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.

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