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**frankmelody** Let X be an infinite dimensional Banach space and show that the unit sphere $\displaystyle S_X=\{||x||=1\}$ is a $\displaystyle G_\delta$ set (i.e. a countable intersection of open sets) in the unit ball $\displaystyle B_X=\{||x||\leq1\}$, 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 $\displaystyle G_\delta$? 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!