Functional Analysis

Prerequisites: 

Measure Theory and Real Analysis

Course description:

This is a course on functional analysis for undergraduate and graduate students. In this course, we will study infinite-dimensional vector spaces and the algebraic, geometric, and analytic properties of operators acting on them. The course covers fundamental concepts, including topological vector spaces, normed linear spaces, Banach and Hilbert spaces, as well as the theory of bounded linear operators. Key theorems such as the Hahn-Banach theorem will be presented, along with the spectral theory of compact and self-adjoint operators. Applications in partial differential equations will also be discussed. The goal of the course is to help students acquire a solid foundation in functional analysis.

Learning objectives:

Through this course, students will gain a theoretical and applied understanding of functional analysis, laying a solid foundation for future scientific research. The curriculum begins with the fundamentals of metric, linear, and topological vector spaces. It then progresses to the study of bounded linear operators, including their spectral theory, eigenvalues, and inverses. A significant portion is dedicated to the properties of Hilbert spaces and the spectral decomposition of self-adjoint operators. The course concludes by exploring applications to partial differential equations.

Detailed topics covered:

1. Metric and linear spaces, basic properties of topological vector space, fixed point theorem, compressed mapping theorem, and implicit function theorem.

2. Normed linear spaces, Banach spaces, convex sets and fixed points, quotient Space, equivalent norms, projection, completeness, compactness.

3. Brouwer and Schauder fixed point theorems.

4. Inner product space, Hilbert space, orthogonality and orthogonal bases, orthogonalization and isomorphism of Hilbert Spaces.

5. Bounded linear operators, Hahn-Banach theorem, Riesz representation theorem and applications, open mapping theorem, closed graph theorem, Lax-Milgram theorem,

6. Continuous linear functional, convex set separation theorem, inversed operators, Banach-Steinhaus Theorem.

7. Conjugate spaces, weak convergence, weak convergence and weak* convergence, reflexive spaces.

8. Eigenvalues, spectral analysis of linear operators, spectral analysis of completely continuous operator, Gelfand Theorem.

9. Basic properties of compact operators, bi-linear functional, spectrum of compact operators, Fredholm operator.

10. Spectral decomposition of self-adjoint operators, applications to partial differential equations.