We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded.

Now, Read's space $R$ (an infinite-dimensional Banach space) has the property: there is $ρ >0$ such that every weakly open subset of the unit ball of $R$ has the diameter greater than or equal to $ρ$.

My question is: since every weakly open subset of an infinite-dimensional Banach vector space X is unbounded then how can a weakly open subset of $R$ be inside the unit ball of $R$?