Thread: unit sphere in an infinite dimensional Banach space

Let X be an infinite dimensional Banach space and show that the unit sphere is a set (i.e. a countable intersection of open sets) in the unit ball , when the latter is endowed with the relative topology inherited from the weak topology of X.

I know that the unit sphere is dense in the unit ball in the above topology, but how to show it is ? And I know that the open set in X under the weak topology of X are all unbounded, and the complement of unit ball is open since unit ball is closed under the weak topology. Thanks in advance!

Originally Posted by frankmelody

Let X be an infinite dimensional Banach space and show that the unit sphere is a set (i.e. a countable intersection of open sets) in the unit ball , when the latter is endowed with the relative topology inherited from the weak topology of X.

I know that the unit sphere is dense in the unit ball in the above topology, but how to show it is ? And I know that the open set in X under the weak topology of X are all unbounded, and the complement of unit ball is open since unit ball is closed under the weak topology. Thanks in advance!

Do you know the theorem that a closed convex subset of a Banach space is weakly closed? That implies that for the set is weakly closed, and so the set is weakly open. Since , it follows that is a set.

If you don't know the "closed and convex implies weakly closed" theorem then you'll need to show from the definition that the set is weakly open. This follows from the Hahn–Banach theorem, which tells you that if then there is a functional f in the unit sphere of the dual space such that .