Hint for problem V.1.9

A function f from X to R is lower semi-continuity if f is continuous into the lower topology on the real line (whose open sets are {x > a}, a real).

One has to show that

if x_i converges to x (in the appropriate weak topology) then liminf ||x_i|| is greater than or equal to ||x||

the closed unit ball is closed in the appropriate weak topology.

Note that the weak-star result can be used in the proof of Proposition 4.1 (instead of Alaoglu's Thm.).

• For the weak topology:
  • Use Corollary 1.5
• For the weak-star topology, the proof is more elementary:
  • Take an element f of the dual whose norm is greater than 1; want to separate it from the closed unit ball, using a weak-star continuous functional.
  • Norm greater than 1 of a functional f means that there is a vector x of norm one such that |f(x)| > 1.
  • Use this vector x to construct the separating functional.