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, 16:30-18:00, Room 005.
Wednesday, 14:30-16:00, Room 005.

Microsoft Teams link



Exams

Syllabus and exam program link

June 16, 10:00, Room 005.
June 30, 10:00, Room 005.
July 14, 10:00, Room 005
July 28, 10:00, ??
September 16, 10:00, ??
September 30, 10:00, ??.



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

    2/03: Introduction to the course. Goals and motivations. How to use the implicit function theorem to solve a nonlinear elliptic PDEs.
    3/03: Bounded linear operators. Adjoints and Hilbert space adjoint. [BS, chap 6] and [RS, chap VI 1,2]
    9/03: Projections in Hilbert spaces. 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].
    16/03: Proof of Fredholm alternative. [B, chap VI].
    17/03: Spectral theory: definitions and topological properties of the spectrum. [BS, chap VII].
    23/03: 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].
    30/03: Continuous functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    31/03: Borelian functional calculus. [RS, chap VII] (see also the first chapter of these lecture notes )
    13/04: Projection valued measure, spectral integral and spectral theorem. [RS, chap VII] (see also the first chapter of these lecture notes )
    14/04: Applications of spectral theorem. [first chapter]
    20/04: Sturm-Liouville problems: from classical to weak solution and back [B, chap VIII.4].
    21/04: 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].
    28/04: Higher order differentiability, Nemitski operator. [AP, chap 1].
    4/05: Implicit function theorem in Banach spaces, Inverse function theorem [AP, chap 2].
    5/05: Applications of implicit function theorem: nonlinear Sturm-Liouville problems, ODE's in Banach spaces and continuous dependence from initial datum [AP, chap 1].
    11/05: Lagrange multipliers in infinite dimensional spaces and complementary spaces [AP, chap 1]. and [B, Section 2.4]
    18/05: Bifurcation theory: Lyapunov-Schmidt reduction. [AP, chap 5].
    19/05: Crandall-Rabinowitz and bifurcation from the simple eigenvalue [AP, chap 5].
    24/05: 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].
    28/05: Topological degree theory: Leray-Schauder degree and Schauder fixed point theorem [T, chap 13].




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. Sturm Liouville operators
  9. Differential Calculus in Banach spaces
  10. Implicit function theorem
  11. ODE's in Banach spaces
  12. Lagrange multipliers
  13. Bifurcationt theory
  14. Stokes wave for Water Waves
  15. Degree theory


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