Locally Convex Vector Spaces

The weak and weak* topologies are introduced for Banach spaces and their duals, and the Banach-Alaoglu theorem is proved. Further topologies are introduced on the spaces of bounded linear operators, and these are placed in the context of the general locally convex vector spaces. Properties of convex bodies are discussed, including the Krein-Milman and Choquet theorems, and several applications of weak* compactness are given. These include Furstenberg’s proof of Weyl’s polynomial equidistribution theorem and elliptic regularity for the Laplace operator at the boundary. The theory of distributions is introduced.