Let $E$ be a Banach space and $E'$ its dual. Show that the weak-* topology on $E'$ is complete. I am trying to prove this statement but am unsure whether the space $E'$ with the given topology is metrizable in the first place. Any help or guidence would be very much appreciated.
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@geetha290krm Hmm, thank you for the answer. So how do we metrize it in the finite dimensional case? – irmbil Apr 29 '25 at 11:54
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1Weak* topology is same as the norm topology if $E$ is finite dimensional. – Kavi Rama Murthy Apr 29 '25 at 11:57
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1@DeanMiller Indeed, it does. Thank you for the answers guys. – irmbil Apr 29 '25 at 11:59
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Sorry, I used the wrong link. The main question is about completeness, not metrisability. Use this link instead as a duplicate. Here is the link addressing metrisability though. – Dean Miller Apr 29 '25 at 12:02