 # Functional and Convex Analysis

### Objective

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.

### Planning

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

### References

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

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

 H. Queffélec. Topologie. Dunod, 2006.

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

 G. Skandalis. Topologie et analyse. Dunod, 2004.

 M. Willem. Analyse fonctionnelle élémentaire. Cassini, 2003.