# A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded

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 $R$?

• Maybe it means weak topology induced on the ball from the whole space? – Mateusz Wasilewski Jan 19 '18 at 6:30
• Could you please give a reference (i.e. a quote) for the statement in your second paragraph? – Yemon Choi Jan 19 '18 at 11:08
• Ref.- arxiv.org/abs/1704.00791 (p. 2) – Infinite Jan 20 '18 at 6:01