Functional Analysis Reading Group

This is the page for the Functional Analysis Reading Group at the University of Missouri, Spring 2020. This twice-weekly seminar is administered by me and Konstantinos Tselios. We will post the schedule and some notes here.

Books:

We will mainly follow Rudin’s outline, with additional materials/proofs from the other books.

N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1963
Walter Rudin, Functional Analysis, McGraw-Hill, 1973
Robert J. Zimmer, Essential Results of Functional Analysis, University of Chicago Press, 1990
Peter D. Lax, Functional Analysis, Wiley, 2002

Schedule:

The basic organization is this: we give an overview of 1-4, with materials of chapter 5 inserted as necessary. Then we’ll proceed with chapter 6-8. If time permits, we will continue with chapter 10, 12, and 13. Exercises are for additional material and can be brought for discussion.
(Note: clicking on the meeting name will bring the lecture notes).
Update: “The Thing” has sadly interrupted normal school schedule and sent us all home, so all talks are cancelled.

  • Week 1:
    • Meeting 1 (27/1, PS): topological vector space (Definition 1.6), types of TVS (Definition 1.8), TVS comes from seminorms (Zimmer Prop. 1.1.4), Examples 1.1.5-10 (Zimmer)
    • Meeting 2 (30/1, PS): base for \(C^\infty\) topology (Theorem 1.37), continuous => bounded (Theorem 1.32), Quotient space (1.40-1.43)
    • Exercise: (Rudin) 1.2, 4, 7, 9, 12, 13,18, 19, 21 (Zimmer) 1.5, 6, 7, 8
  • Week 2:
    • Meeting 3 (3/2, KT): Baire Category (2.1-2.2), Banach-Steinhaus (2.4, 2.6, 2.9)
    • Meeting 4 (6/2, KT): open mapping (2.11, 2.12 (a))
    • Exercise: (Rudin) 2.1, 3, 4, 6, 13
  • Week 3:
    • Meeting 5 (10/2, TH): closed graph (2.14, 2.15)
    • Meeting 6 (13/2, TH): dual space, Riesz-Kakutani (Zimmer A.19), Riesz representation (Akhiezer #16)
    • Exercise: (Rudin) 2.7, 2.8, 2.14
  • Week 4:
    • Meeting 7 (17/2, TH, PS): Riesz representation (cont’d), Hahn-Banach domination (3.2, 3.3)
    • Meeting 8 (20/2, PS): Hahn-Banach separation (3.4, 3.5)

    • Exercise: (Rudin) 3.1, 2, 4 (Zimmer) 2.1
  • Week 5:
    • Meeting 9 (24/2, PS): Hahn-Banach continuous extension (3.6), weak topology (3.11)
    • Meeting 10 (27/2, LF): weak topology (cont’d), with convexity (3.13), weak-* topology (3.14),
    • Exercise: (Rudin) 3.4, 5(a-d), 8, 10 (Zimmer) 1.15, 18, 21, 22, 25
  • Week 6:
    • Meeting 11 (2/3, LF): Example 1.1.27 (Zimmer), weak unit ball is compact (Zimmer 1.1.28), weak => strong convergence (5.1), Banach-Alaoglu (3.15)
    • Meeting 12 (5/3, KT): Theorem 3.19, extreme point (3.20)
    • Exercise: (Rudin) 3.15, 23, 24, 25, 28(c) (Zimmer) 1.11, 13, 16, 19
  • Week 7:
    • Meeting 13 (9/3, KT): Krein-Milman (3.21)
    • Meeting 14 (12/3, PS): Krein-Milman (cont’d), Stone-Weierstrass (5.6, 5.7)
    • Exercise: (Rudin) 3.19, 21 (Zimmer) 2.2
  • Week 8:
    • Meeting 15 (16/3, PS) Kakutani-Markov fixed point (5.11. Zimmer 2.1.5)
    • Meeting 16 (19/3, PS): Haar measure for compact groups (5.14, Zimmer 2.2.1)
    • Exercise: (Rudin) 5.15 (Zimmer) 2.1, 2, 7
  • Week 9:
    • Meeting 17 (30/3, ): dual norm (4.1, 4.3, 4.4), dual of subspace & quotient spaces (4.9), weak and weak-* topology (Zimmer 1.1.25-1.1.30)
    • Meeting 18 (2/4, ): adjoint (4.10, 4.12), compact operator (4.17-19), examples (Zimmer 3.1.4-6, 3.1.11
    • Exercise: (Rudin) 5.6-8, 10-11, 4.1, 4, 6, 7 (Zimmer) 2.3, 5, 6, 11
  • Week 10:
    • Meeting 19 (6/4, ): spectrum of compact operator (Zimmer 3.2.3-8)
    • Meeting 20 (9/4, ): test functions (6.3-5), distribution definition (6.7-9, but compare to Zimmer p. 112-115), basics (6.11-13, 15, 16-17)
    • Exercise:
  • Week 11:
    • Meeting 21 (13/4, ): distribution basics (cont’d), derivative representation (6.26-28), convolution (6.30, 6.33)
    • Meeting 22 (16/4, ): Fourier transform (quick review, Zimmer section 5.1), Plancherel theorem (5.1.11)
    • Exercise:
  • Week 12:
    • Meeting 23 (20/4, ): weak derivative (Zimmer 1.1.17, 18), fractional derivative (Zimmer 5.2.1-3), Sobolev embedding part (I) (Zimmer 5.2.4)
    • Meeting 24 (23/4, ): Sobolev embedding part (I) (cont’d), Corollary 5.2.5-7
    • Exercise:
  • Week 13:
    • Meeting 25 (27/4, ): Sobolev embedding part (I) (cont’d), Corollary 5.2.5-7
    • Meeting 26 (30/4, NU): Rellich embedding (5.2.8)
    • Exercise:
  • Week 14:
    • Leeway

Speakers: