It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is enough to obtain that non-reflexivity.
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3$\begingroup$ This shows that $VMO$ is not reflexive, and a Banach space is reflexive if and only if its dual is. Applying this twice shows that $BMO$ (and $H^1$) are not reflexive either. See here: math.stackexchange.com/questions/152343/… $\endgroup$– Christian RemlingCommented May 2, 2023 at 15:50
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