Building on the basic lectures on Analysis and Linear Algebra, we will study topological vector spaces and linear mappings between them.The crux is hereby that the considered spaces are usually ...
Building on the basic lectures on Analysis and Linear Algebra, we will study topological vector spaces and linear mappings between them.The crux is hereby that the considered spaces are usually infinitely dimensional—think, for instance, at vector spaces consisting of whole functions or sequences. Due to the infinite dimensions, well-known concepts and results from the finitely dimensional theory will fail. However, in many cases, we can establish similar results in a weaker sense. For instance, we will see that every compact set contains a somehow convergent sequence, that some spaces have infinitely dimensional bases, and that certain linear mappings between infinitely dimensional vector spaces possess a spectral decomposition which generalizes the well-known eigenvalue decomposition of matrices. The theoretical concepts of functional analysis are the basis for many other mathematical areas as (partial) differential equations, control theory, nonlinear optimization, and inverse problems.
