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I am reviewing a well established results that in infinite dimensional Banach space $B$ the weak closure of the the unit sphere is the unit ball. I am following a proof from Kesavan's Functional Analysis and I get everything up until he claims that showing every weakly open neighborhood of every point in the open unit ball intersects the unit sphere implies the closed unit ball must lie in the weak closure of the unit sphere. Could someone please elaborate on this part of the proof.

PS: I don't think it is necessary, but I can provide the whole proof if need be.

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It's basic point-set topology. The closure of a set $X$ (in a certain topology, such as the weak topology here) is the intersection of all closed sets containing $X$, or equivalently the complement of the union of all open sets that don't intersect $X$.

If a point $a$ is not in the closure of $X$, that means there's an open set containing $a$ (namely the complement of the closure of $X$) that doesn't intersect $X$. By the contrapositive, if every open set containing $a$ intersects $X$, then $a$ is in the closure of $X$. Done.

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