French version

MASTER ''Mathematical Modelling''

Ecole Polytechnique and Université Pierre et Marie Curie

Master Paris 6 ---------------------- Master Ecole Polytechnique

M2: basic course taught by G. Allaire for the academic year 2015-2016

Functional analysis and applications

This course will be taught in English, see for the French version. It takes place in the Monge amphitheater at Ecole Polytechnique, every Thursday from 09H30 to 12H30 and from 13H30 to 15H30 (from 9/10/2015 to 10/15/2015). To find the Monge amphitheater look at the map on page 3 of this document.

This course is devoted to functional analysis and some applications, including partial differential equations. Each slot features 3 hours of courses and 2 hours of working exercises. At the end of each class (except the first one) a short written exam will take place. The final grade will be a combination of the grades obtained at those short exams.

Contents (A more detailed list of contents is available here.)

  • 1 : Introduction, definition of Banach and Hilbert spaces, definition of L^p Lebesgue spaces.
  • 2 : Linear operators on Banach spaces (Banach-Steinhaus theorem, Banach open mapping theorem, Banach closed graph theorem, orthogonality, unbounded linear operators).
  • 3 : Weak convergence, weak-* convergence, reflexive and separable spaces, application to L^p Lebesgue spaces.
  • 4 : Fredholm theory of compact operators. Spectral decomposition of compact self-adjoint operators on a Hilbert space.
  • 5 : Distributions, convolution, Fourier analysis, Sobolev spaces, embeddings, traces. Applications to partial differential equations.
  • Short bibliography

  • BREZIS H., Analyse fonctionnelle, Masson, Paris (1983).
  • BREZIS H., Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York (2011).
  • CIARLET Ph., Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia (2013).
  • Exercises (by Nicole Spillane, based on earlier notes of Olivier Pantz)

  • Class of September 10, 2015.
  • Class of September 17, 2015.
  • Class of September 24, 2015.
  • Class of October 01, 2015.
  • Class of October 05, 2015.
  • Class of October 15, 2015.
  • Every week the new exercise sheet is posted on this web page. The solutions are posted one week later.

    Written exam

    The written exam for this course will take place at Ecole Polytechnique on Thursday, January 7th, 2016, from 2pm to 5pm. The exam will take place in the seminar room of CMAP, 3rd floor, wing 0 of the laboratories. For access see this web page.