The weak topology of locally convex spaces and the weak-* topology of their duals

These notes give a summary of results that everyone who does work in functional analysis should know about the weak topology on locally convex topological vector spaces and the weak-* topology on their dual spaces. The most striking of the results we prove is Theorem 9, which shows that a subset of a locally convex space is bounded if and only if it is weakly bounded. It is straightforward to prove that if a set is bounded then it is weakly bounded, but to prove that if a set is weakly bounded then it is bounded we use the Hahn-Banach separation theorem, the Banach-Alaoglu theorem, and the uniform boundedness principle. If X is a topological vector space then we will see that the weak topology on it is coarser than the original topology: any set that is open in the original topology is open in the weak topology. From this it follows that it is easier for a sequence to converge in the weak topology than in the original topology: for a sequence to converge to a point means that it is eventually contained in every neighborhood of the point, and a point has fewer neighborhoods in the weak topology than it does in the original topology. The weak topology encodes information we may care about, and we may be able to establish that certain sets are compact in the weak topology that are not compact in the original topology. In these notes I first define the weak topology on a topological vector space X, and show that if X is locally convex then X with the weak topology is also a locally convex space. Indeed a normed space is locally convex, but there are function spaces that we care about that are not normed spaces. For example, the set of holomorphic functions on the open unit disc is a Fréchet space that is not normable. If U is an open subset of R, then C(U) is a Fréchet space that