Math 205 - Analysis II - Winter 2018

       

General info

• Syllabus
• Lectures: MW 11:40-1:15pm, McHenry 1270.
• Office hours: Mondays 3-4pm, Wednesdays 2-3pm.
• Main reference: "Real Analysis" by Stein and Shakarchi [SS]
• Other good references:
  • "Real and Complex Analysis" by W. Rudin [Rudin]
  • "Real Analysis", by Royden and Fitzpatrick [Royden]
  • "Functional Analysis", by Brezis [Brezis]

Lectures

• 1/8 - Lecture 1: Preliminaries. The Exterior Measure. [SS, Ch. 1, Sec 1-2] Outline
• 1/10 - Lecture 2: Properties of the Exterior Measure. Measurable sets, the Lebesgue measure. [SS, Ch. 1, Sec 2-3] Outline
• 1/17 - Lecture 3: Measurable functions. [SS, Ch. 1, Sec 3-4] Outline
• 1/18 - Lecture 4: Approximation of measurable functions. Egorov's and Lusin's theorems. [SS, Ch. 1, Sec 4] Outline
• 1/22 - Lecture 5: Construction of the Lebesgue integral, convergence theorems. [SS, Ch. 2, Sec 1] Outline
• 1/24 - Lecture 6: The Lebesgue integral (continued). The Banach space L1. [SS, Ch. 2, Sec 1-2] Outline
• 1/29 - Lecture 7: The Banach space L1. Theorems of Fubini and Tonelli. [SS, Ch. 2, Sec 2-3] Outline
• 2/5 - Lecture 8: Differentiation of the integral. Lebesgue differentiation theorem. [SS, Ch. 3, Sec 1-2] Outline
• 2/7 - Lecture 9: Integration of the derivative. Bounded variation. Absolute continuity. [SS, Ch. 3, Sec 3] Outline
• 2/12 - Lecture 10: Hilbert spaces. The space L2. Riesz representation. [SS, Ch. 4, Sec 1,2,5] Outline
• 2/14 - Lecture 11: Lp spaces. [Royden, Ch. 3,4][Brezis, Ch. 7,8] Outline
• 2/21 - Lecture 12: Abstract measure spaces. [SS, Ch 6, Sec 1] Outline
• 2/26 - Lecture 13: Extension theorem. Integration on Measure spaces. Product measures and Fubini. [SS, Ch 6] Outline
• 2/28 - Lecture 14: Completeness, Fubini, the Lebesgue-Stieltjes integral. [SS, Ch 6] Outline
• 3/5 - Lecture 15: Signed measures, the Lebesgue-Radon-Nikodym theorem. [SS, Ch 6] Outline
• 3/7 - Lecture 16: Hausdorff measures and fractals. [SS, Ch 7] Outline
• 3/12 - Lecture 17: The Spectral Theorem 1/2. Outline
• 3/14 - Lecture 18: The Spectral Theorem 2/2. Outline

Problem sets

• 1/9 - HW1 - due 1/17.
• 1/17 - HW2 - due 1/24.
• 1/24 - HW3 - due 2/7 (due date updated).
• 2/8 - HW4 - due 2/21 (due date updated).
• 2/21 - HW5 - due 2/28.
• 2/28 - HW6 - due 3/07.
• 3/8 - HW7 - due 3/14.