I'm learning about the weak and weak* topologies on a normed vector space following the book of Brezis. He limits his discussion to case where $E$ is a Banach space, and my question is most simply stated as, "Why?". I can't find an example of a theorem where the completeness of $E$ is a necessary hypothesis. Most of the basic results on weak and weak* topologies proceed by applications of Hahn-Banach, which holds for a much wider class of spaces than just Banach spaces.
Are there any examples of reasonably elementary (i.e. of relevance to a first-year graduate student who does not anticipate having heavy contact with functional analysis in the future) facts about weak or weak* topologies that are true for Banach spaces but not all normed vector spaces?
EDIT: I should add that, as Yemon Choi points out, dual spaces of normed vector spaces are complete, so the weak-star topology will never be defined on a space that isn't Banach. With regards to the weak-star topology, then, my question should refer to aspects of the weak-star topology on a space $E^*$ where the original $E$ is not Banach.