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Analysis II, Spring 2021

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Course Information

• Ceiba (link retired)
• Lectures: Tuesday, 10:20 ~ 12:10 at Astro-Math 101
  Thursday 10:20 ~ 12:10 at Astro-Math 101
• Tuesday 14:00 ~ 15:00, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have two jokers: the lowest two grades will be discarded.
  • Quiz 5% (on March 16)
  • Midterm 30% (on April 20)
  • Final Exam 35% (on June 15)
• Course prerequisite:
  • freshman calculus (for math major)
  • linear algebra (for math major)
  • analysis I
• Textbook:
  • Elias M. Stein and Rami Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces. MR
  • Glen E. Bredon, Topology and Geometry. MR
• 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) topological space, subspace, connectivity, separation axiom, compactness. Reference: Bredon §1.2 ~ §1.7 except §1.6
• (Week 2) product topology, Urysohn metrization theorem, one-point compactification, locally compact. Reference: Bredon §1.8 ~ §1.11
• (Week 3) paracompact, partition of unity, quotient topology. Reference: Bredon §1.12 ~ §1.13
  Lebesgue measure. Reference: Stein §1.2, §1.3
• (Week 4) measurable function, Brunn--Minkowski inequality, Lebesgue integral via simple functions. Reference: Stein §1.4, §1.5, §2.1
• (Week 5) Lebesgue integral (continued), space of integrable functions, Fubini theorem, Fourier transform of integrable functions. Reference: Stein §2.2, §2.3, §2.4
• (Week 6) Fourier inversion (continued), Hardy-Littlewood maximal function. Reference: Stein §2.4, §3.1
• (Week 7) Lebesgue differentiation theorem, approximations to the identity. Reference: Stein §3.1, §3.2
• (Week 8) BV function, rectifiable curve, Minkowski content. Reference: Stein §3.3, §3.4
• (Week 9) isoperimetric inequality, Hilbert space, orthonormal basis. Reference: Stein §3.4, §4.1, §4.2
• (Week 10) Spring Break.
• (Week 11) Poisson kernel, Fourier series of L¹ and L² functions, closed subspace, orthogonal projection, Riesz representation theorem, adjoint transform. Reference: Stein §4.3, §4.4, §4.5
• (Week 12) integral operator, Hilbert--Schmidt operator, compact operator, spectral theorem of compact, self-adjoint operator. Reference: Stein §4.5
• (Week 13) abstract measure, outer measure, Carathéodory measurability, metric outer measure, premeasure, Carathéodory extension, integration. Reference: Stein §6.1, §6.2
• (Week 14) product measure, Fubini theorem, Riemann--Stieltjes integral, signed measure, absolute continuity of measures. Reference: Stein §6.3, §6.4
• (Week 15) Radon--Nikodym theorem, two applications in probability theory, mean ergodic theorem, maximal ergodic theorem. Reference: Stein §6.4, §6.5
• (Week 16) pointwise ergodic theorem, ergodic measure-preserving transformation, unique ergodicity, mixing, Hausdorff measure, Hausdorff dimension, Holder continuity, Cantor set. Reference: Stein §6.5, §7.1, §7.2

Homework

• Homework 01 (due March 2) Bredon. p.7 #2, p.9 #3, p.10 #5, #6, #7, p.12 #4, p.14 #4.
• Homework 02 (due March 9) Bredon. p.22 #3, p.24 #3, #4, #5, #6, p.31 #1.
• Homework 03 (due March 16) Bredon. p.43 #2, p.44 #6, #7.
• Homework 04 (due March 23) Stein. ch.1 exercise #7, #8, #19, #26, #35, #38.
• Homework 05 (due March 30) Stein. ch.2 exercise #5, #6, #10, #12, #13, #15.
• Homework 06 (due April 8) Stein. ch.2 exercise #9, #21, #22, #24.
• Homework 07 (due April 13) Stein. ch.3 exercise #4, #5.
• Homework 08 (due May 4) Stein. ch.2 exercise #23, #25. ch.4 exercise #4, #5.
• Homework 09 (due May 11) Stein. ch.4 exercise #7, #8, #10, #11, #13, #22.
• Homework 10 (due May 18) Stein. ch.4 exercise #18, #20, #21, #25.
• Homework 11 (due May 25) Stein. ch.4 exercise #28, #29, #32. ch.6 exercise #2, #3.
• Homework 12 (due June 1) Stein. ch.6 exercise #5, #12, #15, #16.
• Homework 13 (due June 8) Stein. ch.6 exercise #9, #10, #18. problem #5.