Analyse Fonctionnelle
Aperçu des sections
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This course will introduce students to the fundamental concepts and tools of functional analysis, a branch of mathematics that studies infinite dimensional vector spaces and the linear operators acting on them. The course will emphasize the applications of functional analysis to probability theory and statistics.
Course Content
Chapter 1: Basic Concepts of Functional Analysis
1.1 Linear Spaces: This section will cover the definition of linear spaces, subspaces, linear independence, spanning sets, basis, and dimension.
1.2 Banach Spaces: This section will introduce the concept of normed linear spaces and Banach spaces. It will discuss completeness, convergence of sequences and series, and examples of Banach spaces such as Lp spaces.
1.3 Spaces of Bounded Operators: This section will explore the properties of linear operators between Banach spaces. It will cover bounded operators, operator norms, and the space of bounded linear operators as a Banach space itself.
Chapter 2: Dual Spaces
2.1 Hahn-Banach Theorem: This section will present the Hahn-Banach theorem, a fundamental result in functional analysis that guarantees the existence of continuous linear functionals extending given functionals defined on subspaces.
2.2 Linear Forms in Banach Spaces: This section will delve into the study of continuous linear functionals on Banach spaces, also known as elements of the dual space. It will cover the properties of the dual space and examples like the dual of Lp spaces.
2.3 Dual of an Operator: This section will introduce the concept of the adjoint operator and explore its connection to the dual space. It will discuss properties of the adjoint and its role in studying operators.
Chapter 3: Semigroups of Operators
3.1 Banach-Steinhaus Theorem: This section will present the Banach-Steinhaus theorem, also known as the uniform boundedness principle, which provides a criterion for uniform boundedness of a family of bounded linear operators.
3.2 Closed Operators: This section will introduce the concept of closed operators, which are more general than bounded operators and arise naturally in many applications. It will discuss properties of closed operators and their relationship to bounded operators.
3.3 Semigroups of Operators: This section will explore the theory of semigroups of operators, which are families of operators satisfying certain algebraic and topological properties. It will discuss the generation of semigroups, their infinitesimal generators, and their connection to evolution equations.
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