a direct consequence of the hahn banach theorem is the if $L\subset X$ is a closed subspace, then $(L^{\perp})_{\perp}=L$. what I'm trying to understand is whether or not for every $L\subset X^*$ closed we similarly get that $(L_{\perp})^{\perp}=L$. I get a strong indication from the book I'm reading that that is not the case, but I can't seem to find a counterexample.
2026-03-25 23:36:17.1774481777
taking the orthogonal complement twice in a banach space
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