MAT 1001 / MAT 458 Real Analysis II, Winter 2022

Tentative Schedule:

Week 1 (January 10-14)
L^p -spaces (Folland Chapter 6; Lieb-Loss Chapter 2 Sections 1-8, 14-16, 20)

W: Motivation and overview
      Hölder and Minkowski inequalities; Hanner's inequality     Recording (Passcode 9Mb2BCjH@9)

F: Proof of Hanner's inequality. Projection onto closed convex sets     Recording (Passcode $uYm$pC5w8)

Assignment 1 (due January 18)      Comments and corrections

Week 2 (January 17-21)
L^p -spaces (cont'd)

W: Dual characterization of L^p -norms
      Riesz representation theorem     Recording (Passcode aEH3XQ@2g1)

F: Proof of Riesz' represenation theorem (conclusion)     Recording (Passcode VUtfHt52t#)

Assignment 2 (due January 25)      Comments and corrections

Week 3 (January 24-28)
L^p -spaces (conclusion.)

W: The scale of L^p -spaces
Complex interpolation. Statement of the Riesz-Thorin theorem     Recording (Passcode =@54GL^8.9)

F: Three-line lemma, and proof of the Riesz-Thorin theorem     Recording (Passcode 37JC#8+zRc)

Assignment 3 (due February 1)      Comments and corrections

Week 4 (January 31-February 4)
Abstract functional analysis (Folland Chapter 5; Brézis Chapter 1; Lieb-Loss Chapter 2 Sections 9-13 and 17-18; Royden 10.2 and Chapters 13-15)

W: Hahn-Banach theorem
     Linear transformations; dual spaces     Recording (Passcode jEz9jQA6+v)

F: Continuous linear functionals and closed hyperplanes     Recording (Passcode 5uh7tC+i$u)

Assignment 4 (due February 8)

Week 5 (February 7-11)
Abstract functional analysis (cont'd)

W: Separation of convex sets
      Legendre transform     Recording (Passcode !%.3p6BrE8)

F: Fenchel-Moreau theorem. Weak and weak-* topologies     Recording (Passcode %38?SnxHEu)

Assignment 5 (due February 15)      Comments and corrections

Week 6 (February 14-18)
Abstract functional analysis (cont'd)

W: Baire category theorem
      uniform boundedness principle, open-mapping theorem, linear homeomorphisms     Recording (Passcode uqpg*8Y9UB)

F: Closed-graph theorem. Weak compactness: Banach-Alaoglu theorem (initial discussion)     Recording (Passcode QxL6t?@1eP)

Assignment 6 (due March 1)      Comments and corrections

Reading week (February 21-25)

Back to the classroom !
Week 7 (February 28 - March 4)
Hilbert spaces (Folland Section 5.5; Stein-Shakarchi Chapter 4; Lieb-Loss Section 2.21)

W: More about weak topologies. Banach-Alaoglu theorem
      Hilbert spaces: orthonormal bases; separability       Recording (Passcode rM6V4bEEq#) Note: First hour is incomplete

F: The adjoint of an operator. Self-adjoint operators; orthogonal projections     Recording (Passcode rYA2y7H.#?)

Old tests: 2013 Problems 3 4 (1b is false), 2014 Problems 1 3 4, 2020 Problems 1a 3 4 (1b is false)

Week 8 (March 7-11)
Hilbert spaces (cont'd)

W: Midterm test: 2022 (during lecture time)

F: Spectral theorem for compact self-adjoint operators     Recording (Passcode ?FWBxfu3$4)

Assignment 7 (due March 15)      Comments and corrections

Week 9 (March 14-18)
Fourier integral transform (Folland Chapter 8; Lieb-Loss Chapter 5; Stein-Shakarchi Sections 2.4 and 5.1)

W: Proof of the Spectral Mapping Theorem
      The Fourier transform on L¹. Translation, rotation, and scaling properties       Recording (Passcode H3ACbown$P)

F: Extension to L². Plancherel's identity

Assignment 8 (due March 22)      Comments and corrections

Week 10 (March 21-25)
Fourier integral transform (cont'd)

W: Fourier inversion
      Schwartz space; the Sobolev spaces H^s       Recording (Passcode 1inY5gM+gv)

F: Distributions. The space of test functions, its topology, and its dual     Recording (Passcode @4U?9DEE7c)

Assignment 9 (due March 29)      Comments and corrections

Week 11 (March 28-April 1)
Distributions (Folland Chapter 9; Lieb-Loss Chapter 6; Stein-Shakarchi Sections 2.4 and 5.1)

W: Weak convergence, weak derivatives
      The Sobolev spaces W^k,p       Recording (Passcode 2%Ms7M3P=B)

F: Convolutions; smooth approximation     Recording (Passcode #BqzpYY4=X)

Assignment 10 (due March 29)

Week 12 (April 4-8)
Some applications of distributions to PDE

W: Weyl's lemma. Poisson problems
      Dirichlet's principle. The direct method       Recording (Passcode P0^2+yr@p#)

F: Weak solutions of the continuity equation     Recording (Passcode 7RMV6Wv%kC)

Old exams: 2014 (in 5c use the Sobolev inequality ||u||_n/(n-1) ≤ C ||∇u||₁; ignore Problem 6),
          2015 (ignore Problem 1d),  2020 (in 6a use the Sobolev inequality ||u||₆ ≤ C ||∇u||₂ )

Final exam April 11, 2-5pm (in-person; St. Hilda's College, Cartwright Lecture Hall)

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