Functional analysis

Aim of the course is to introduce the basic tools of linear and nonlinear analysis, and to apply them in analysis and mathematical physics. The course is divided in two parts: Part 1 covers the spectral theory of linear operators. Part 2 introduces the tools of infinite dimensional calculus and nonlinear methods.

Sissa page of the course link



Timetable


Monday 09:15-10:45, Thursday 09:15-10:45

Microsoft Teams link





Main Textbooks

  • [BS] Bogachev, Smolyanov: Real and Functional Analysis. Moscow Lectures, Springer 2020, link
  • [B] Brezis, Functional analysis, Springer 2011, link
  • [EMT] Eidelman, Milman, Tsolomitis. Functional analysis.Graduate studies in Mathematics, 66. American Mathematical Society, 2004
  • [RS] Reed, Simon: Methods of modern mathematical physics. I. Functional analysis. Academic Press, Inc., New York, 1980
  • [AP] Ambrosetti, Prodi: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, 1995
  • [C] Chang: Methods in nonlinear analysis. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005
  • [T] Teschl: Topics in Linear and Nonlinear Functional Analysis, Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, link


Diary

    3/03: Introduction to the course. Projections in Hilbert spaces.
    6/03: Convergence of operators. Neumann series. Compact operators and their main properties. [BS, chap 6] and [B, chap VI].
    10/03: Examples of compact operators. Fredholm theory: statement and preliminary results. [BS, chap 6] and [B, chap VI].
    13/03: Proof of Fredholm alternative. [B, chap VI].
    17/03: Spectral theory: definitions and topological properties of the spectrum. [BS, chap VII].
    20/03: Spectral theory of selfadjoint operators. [BS, chap VII] and [B, chap VI].
    24/03: Spectral theory of compact operators. [BS, chap VII]. and [B, chap VI].
    27/03: Continuous functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    31/04: Riez-Markov theorem, Borelian functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    3/04: Projection valued measures, spectral integral. [RS, chap VII] (see also the first chapter of these lecture notes )
    7/04: Applications of spectral theorem [Chap. 5]
    1/04: Quantum dynamics




Lecture notes

  1. Projections, Neumann series
  2. Compact operators
  3. Fredholm theory
  4. Spectral theory
  5. Spectrum of Compact Operators
  6. Functional calculus
  7. Spectral theorem
  8. Quantum Dynamics
  9. Sturm Liouville operators
  10. Differential Calculus in Banach spaces
  11. Implicit function theorem
  12. Periodic orbits
  13. ODE's in Banach spaces
  14. Lagrange multipliers
  15. Bifurcationt theory
  16. Stokes wave for Water Waves
  17. Degree theory


Exercises

  1. Sheet 1
  2. Sheet 2
  3. Sheet 3


Additional material

  1. Spectral theory examples
  3. A lemma in measure theory useful for the Borellian functional calculus


Lecture notes

  • [CR] Cheverry, Raymond: Handbook of spectral theorem link
  • [P] Pankrashkin: Introduction to spectral theory link
  • [W] Williams: Lecture notes on the spectral theorem link


Other Textbooks

  • [AP] Ambrosetti, Prodi: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, 1995
  • [C] Chang: Methods in nonlinear analysis. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005
  • [Ki] Kielhöfer: Bifurcation theory, Springer, 2004, link
  • Kreyszig: Introductory Functional Analysis with applications link
  • [L] Lax: Functional analysis, Wiley, 2002.
  • [LB] Levy-Bruhl: Introduction à la théorie spectrale, Dunod, 2003
© Tetiana Savitska 2017