Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of $(E^*)^d$ with finite codimension. I would like know if the space $\mathcal G$ is a dual space $\mathcal E^*$ and if there is a somewhat explicit and canonical description of $\mathcal E.$
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$\begingroup$ By "weakly closed" do you mean "weak-star closed"? $\endgroup$– Yemon ChoiCommented Mar 12, 2015 at 15:51
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$\begingroup$ Because if you mean weak-star closed then the answer is yes (without any assumption on codimension). On the other hand, norm closed subspaces are weakly closed. $\endgroup$– Yemon ChoiCommented Mar 12, 2015 at 17:05
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$\begingroup$ Yes, I mean weak-star closed. $\endgroup$– BazinCommented Mar 12, 2015 at 21:04
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Recall the canonical isometric isomorphism $ (X/N )^*\sim N^\perp$, for a Banach space $X$, and a closed linear subspace $N$ of $X $. Also, a linear subspace $G$ of $X^*$ is weakly-star closed if and only if $G=(G_\perp)^\perp$. So in the latter case $ G\sim (X/G_\perp )^*$.
answered Mar 13, 2015 at 3:11