The purpose of this course is to deepen some notions of functional analysis that can be encountered in the study of mathematical models of biological problems. For instance :
- Banach spaces. The notion of norm on a vector space. Convergent sequences and notion of Banach space. Fixed point Theorem.
- Examples in finite dimension. Completeness. Notion of compactness and the link with closed and bounded subsets. Illustrations in biology : problems of optima. Behavior of recursive sequences. Fixed points of discrete dynamical systems. Stability/instability of a fixed point.
- Examples in infinite dimension. Examples of classical functional spaces. Spaces of continuous functions on a compact set, C(K,Rn), H1([0,1]), spaces of periodic functions. Linear continuous operators.
- Examples of applications of the fixed point theorem. Illustrations in Biology.
- Study of functional equations, integral equations. Examples from Biology.
- Study of density and approximation property (Dirichlet Theorem, Gibbs phenomenon, Theorems of Fejer and of Weierstrass.)