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


Wednesday, 14:30-16:00, Room 128. March 16, Room 136. From May 4, Room 005.

Thursday, 14:30-16:00, Room 005.
Microsoft Teams link



Exams

Syllabus and exam program link

June 10, 10:00, Room 005.
June 27, 14:00, Room 005.
July 22, 10:00, Room 005



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
  • [N] Nirenberg: Topics in Nonlinear Functional Analysis, Courant Lecture Notes, AMS.


Diary

    2/03: Introduction to the course. Goals and motivations. How to use the implicit function theorem to solve a nonlinear elliptic PDEs.
    3/03: Projections in Hilbert spaces. Convergence of operators. Neumann series. Compact operators and their main properties. [BS, chap 6] and [B, chap VI].
    8/03: Examples of compact operators. Fredholm theory: statement and preliminary results. [BS, chap 6] and [B, chap VI].
    9/03: Proof of Fredholm alternative. [B, chap VI].
    16/03: Spectral theory: definitions and topological properties of the spectrum. [BS, chap VII].
    17/03: Spectral theory of selfadjoint operators. [BS, chap VII] and [B, chap VI].
    23/03: Spectral theory of compact operators. [BS, chap VII]. and [B, chap VI].
    24/03: Continuous functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    6/04: Borelian functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    7/04: Projection valued measure, spectral integral and spectral theorem. [RS, chap VII] (see also the first chapter of these lecture notes )
    13/04: Applications of spectral theorem, Quantum dynamics [Chap. 5]
    21/04: Sturm-Liouville problems: from classical to weak solution and back [B, chap VIII.4].
    26/04: Applications of Sturm-Liouville theory. [B, chap VIII.4].
    27/04: Differential calculus in Banach spaces: Frechet and Gateux differentials, relations and examples. [AP, chap 1].
    04/05: Higher order differentiability, Nemitski operator. [AP, chap 1].
    05/05: Implicit function theorem in Banach spaces, Inverse function theorem [AP, chap 2].
    06/05: Applications of implicit function theorem: nonlinear Sturm-Liouville problems, periodic solutions to ODE's [AP, chap 1].
    11/05: Lagrange multipliers in infinite dimensional spaces and complementary spaces [AP, chap 1]. and [B, Section 2.4]
    12/05: Bifurcation theory: Lyapunov-Schmidt reduction. [AP, chap 5].
    18/05: Crandall-Rabinowitz and bifurcation from the simple eigenvalue [AP, chap 5].
    19/05: Construction of the Stokes wave for water waves [AP, chap 5].
    20/05: Shauder's fixed point theorem [N, chap 2].
    31/05: Exercises correction: part 1
    31/05: Exercises correction: part 2




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. Shauder's fixed point theorem


Exercises

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


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