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


Tuesday, 9:15-11:00, Room 128. May 02, Room 136.

Wednesday, 9:15-11:00, Room 128. May 03, Room 136.
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

    02/05: Sturm-Liouville problems: from classical to weak solution and back [B, chap VIII.4].
    03/05: Applications of Sturm-Liouville theory. [B, chap VIII.4].
    9/05: Differential calculus in Banach spaces: Frechet and Gateux differentials, relations and examples. [AP, chap 1].
    10/05: Higher order differentiability, Nemitski operator. [AP, chap 1].
    16/05: Implicit function theorem in Banach spaces, Inverse function theorem [AP, chap 2].
    17/05: Applications of implicit function theorem: nonlinear Sturm-Liouville problems, periodic solutions to ODE's [AP, chap 1].
    23/05, I: Lagrange multipliers in infinite dimensional spaces and complementary spaces [AP, chap 1]. and [B, Section 2.4]
    23/05, II: Bifurcation theory: Lyapunov-Schmidt reduction. [AP, chap 5].
    24/05, I: Crandall-Rabinowitz and bifurcation from the simple eigenvalue [AP, chap 5].
    24/05, II: Construction of the Stokes wave for water waves [AP, chap 5].
    25/05: Topological degree theory: Brouwer degree and Brouwer fixed point theorem [T, chap 12].




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
  4. Sheet 4


Exams

  1. June 2023


Additional material

  1. Spectral theory examples


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