2

I am confused by the answer which says “apply Banach-Alaoglu to the unit ball of $H^{**}$”. I can’t see how to get the result we desired by this. Does it means

Since $(x_n)$ is a bounded sequence in $H^{**}$, it has a convergent subsequence $(x_{n_k})$ on the $w^{*}$-compact unit ball of $H^{*}$, so $x_{n_k}\rightharpoonup x_n$?

Any help would be appreciated.

mio
  • 1,048
  • 1
  • 4
  • 14
  • 1
    I might be wrong but I think you need the Eberlein-Smulian theorem ( https://en.wikipedia.org/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem ). This is because the implication of compactness to sequential compactness holds for I-countable spaces which is not the case in general for dual spaces with the weak topology. – MIO Apr 26 '23 at 10:39
  • 1
    Thanks for your comment. – mio Apr 26 '23 at 11:27

0 Answers0