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.

Assessment:

The overall grade for the course will be the average of the continuous assessment (“contrôle continu”, CC) grade (50%) and the written final exam (50%).

The continuous assessment grade is made up of three elements, each graded out of twenty points: (i) the mid-term grade, (ii) the grade for attendance at tutorial sessions (TD), whose attendance is mandatory, and (iii) the grade for participation in tutorial sessions. The CC grade is calculated as follows: 50% of the mid-term grade + 25% of the attendance grade + 25% of the maximum between the participation grade and the mid-term grade.

The attendance grade is calculated according to the scale available on the school's intranet.

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

- Boyd et Vandenberghe, Convex optimization

- H. Brézis, Analyse Fonctionnelle

- K.Kuttler, Modern Analysis

- Meise et Vogt, Introduction to Functional Analysis

- W. Rudin, Analyse Réelle et Complexe