Functional Analysis Review

These notes are organized as follows. We first discuss finite dimensional vector spaces and additional structures that we can impose on them. We then introduce Hilbert spaces, which play a central role in this class, and talk about the subtleties that we have to deal with when working in infinite dimension. Finally, we talk about matrices and linear operators, in the context of linear mappings between vector spaces. We will see how matrices represent linear functions between finite dimensional vector spaces, and develop a parallel theory on linear operators between general Hilbert spaces.