ABSTRACT
In this chapter, the authors study continuous linear functionals on topological vector spaces. They begin with a brief treatment of continuous linear functionals. From here the authors develop the structure of the (continuous) dual space. They show that the dual space is a vector space. Next, the authors examine some topologies that can be placed on the dual space, such as the weak, strong, and inductive limit topologies. They develop a few results about linear functionals on a nuclear space. The authors also develop a version of one of the most important results in all of functional analysis, the Banach–Steinhaus Theorem and present multiple versions of the Hahn–Banach Theorem. The Banach–Alaoglu Theorem tells that the closed unit ball is compact in the weak topology.
