Syllabus
An introductory course to functional analysis focused on the extension of the results of linear algebra to infinite-dimensional spaces and on applications of general theoretical results within partial differential equations.
1. Introduction Finite-dimensional vector spaces and linear representations (summary). Function spaces, metric spaces, normed spaces. Banach and Hilbert spaces. Compactness in finite-dimensional and infinite-dimensional spaces.
2. Linear operators Continuous linear operators, examples. Hahn-Banach theorem and its consequences. Dual spaces, weak and weak-* convergence. Reflexive spaces. Banach-Alaoglu theorem.
3. Bounded linear operators Principle of uniform boundedness, open mapping theorem and closed graph theorem. Adjoint operator, compact operator.
4. Hilbert spaces Orthogonal projections, Riesz representation theorem. Lax-Milgram lemma and its application in the theory of partial differential equations. Introduction to Sobolev spaces. Compact operators. Fredholm alternative. Spectrum. Self-adjoint operators, Hilbert-Schmidt theorem.
Annotation
A basic course in functional analysis focusing on applications of general theory in the context of the theory of partial differential equations.