TITLE: Norm is weakly lower semicontinuous
QUESTION [17 upvotes]: Is it the case that for every normed space, the norm is always weakly lower semicontinuous? Does it also hold for topologies other than the weak one?

REPLY [5 votes]: Let X be a normed space. We want to show that, for any λ≥0, the set {x∈X:∥x∥≤λ} is weakly closed.Firstly,the set {x∈X:∥x∥≤λ} is convex. Secondly, {x∈X:∥x∥≤λ} is closed in the norm topology.Hence,the result follows from the fact that a convex set's closure is the closure in the weak topology.