ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Functional and Convex Analysis


This course is designed to teach first-year students the essential aspects of topology and convex and non-linear analysis (fixed point theorems) to prepare them for later mathematics and economics studies.



  1. Further topology -Metric, compact and locally compact spaces, complete metric spaces, extension of continuous functions with applications, Banach's fixed point theorem, Baire's theorem and its applications, connected spaces.
  2. Functional spaces -The Stone-Weierstrass theorem, Ascoli's theorem.
  3. Normed vector spaces -Continuous linear and multilinear maps, topological duals of a normed space, strong topology, the Hahn-Banach theorem, the bidual of a normed space, reflexivity.
  4. Banach spaces -The Banach-Steinhaus theorem, the open map and closed graph theorems.
  5. Convexity -Convex sets, convex envelope, cones, topological properties of convex sets, convex (concave) functions and their topological properties, theorems of separation, polarity and orthogonality, Farkas' lemma, quasi-convex (quasi-concave) functions
  6. Real Hilbert spaces -Best approximation projector, conic and linear projectors, duality in Hilbert spaces and the adjoint of a continuous operator.
  7. Fixed point theorems -Brouwer's fixed point theorem, extension to Hilbert spaces, ideas on correspondences, Kakutani's theorem, extension to Banach spaces, the Debreu-Gale-Nikaido lemma


[1] J.B. Hiriart-Urruty et C. Lemaréchal. Fundamentals of Convex Analysis. Springer, 2001.

[2] F. Hirsch et G. Lacombe. Eléments d’analyse fonctionnelle. Dunod, 1997.

[3] H. Queffélec. Topologie. Dunod, 2006.

[4] L. Schwarz. Analyse I. Théorie des ensembles et topologie. Hermann, 1997.

[5] G. Skandalis. Topologie et analyse. Dunod, 2004.

[6] M. Willem. Analyse fonctionnelle élémentaire. Cassini, 2003.