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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 $R$?

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    $\begingroup$ Maybe it means weak topology induced on the ball from the whole space? $\endgroup$ – Mateusz Wasilewski Jan 19 '18 at 6:30
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    $\begingroup$ Could you please give a reference (i.e. a quote) for the statement in your second paragraph? $\endgroup$ – Yemon Choi Jan 19 '18 at 11:08
  • $\begingroup$ Ref.- arxiv.org/abs/1704.00791 (p. 2) $\endgroup$ – Infinite Jan 20 '18 at 6:01
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I think that it should be as Mateusz Wasilewski wrote, that every relatively weakly nonempty open subset of the unit ball of Read's space has diameter greater than or equal to 2/3. The reference is Corollary 8 in here: https://arxiv.org/abs/1704.00791

So to answer the original question, then relatively weakly open sets of the unit ball are intersections of weakly open sets with the unit ball, which is bounded. Hence, relatively weakly open sets are bounded too.

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  • $\begingroup$ Nice detective work. $\endgroup$ – Nik Weaver Jan 19 '18 at 19:47
  • $\begingroup$ Thanks! I have done some research in that direction, that is why I knew the paper. $\endgroup$ – Johann Langemets Jan 20 '18 at 7:20

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