Let $E$ be a reflexive Banach space and $B \subset E$ be a closed Banach subspace. I'm trying to show that $B$ is also a reflexive space. To this end, we can take $(x_n)_n \subset B$ a bounded sequence and show that $(x_n)_n$ has a subsequence that converges weakly. If $\ell \in B^*$, then $\ell$ can be extended to $\tilde{\ell} \in E^*$ such that $\tilde{\ell}|_{B} = \ell$ and therefore, there exists $x \in E$ with $x_{n_k} \rightharpoonup x$ for $(x_{n_k})_k \subset (x_n)_n$. But now I do not really know how to conclude that $x \in B$. Is it the right way to do that ? Does anyone know how to conclude ?
2026-02-23 10:04:44.1771841084
A closed Banach subspace of a reflexive Banach space is also reflexive.
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