TrainingsMasterMathématiques appliquées, statistiqueCoursesFunctional analysis

Master Mathématiques appliquées, statistiqueUE Functional analysis

Content

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.)

Language used

Main language used by this course: Anglais.

Structure and organisation

Lectures will provide a summary of basic concepts, which will be applied in practicals.

Volume of teachings

  • Lectures: 9 hours
  • Tutorials: 9 hours

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