A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the unit ball in a space $Y$.

Now my question is if a related result holds when $X$ is only a quasi-Banach space, say if we impose some richness conditions for the dual spaces.

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    $\begingroup$ I suggest that you think about $\ell^p$ with $0<p<1$. $\endgroup$ Sep 7, 2023 at 23:31
  • $\begingroup$ First of all, sorry for the late reply! The argument should be: for any $p \leq 1$, $(\ell^p)^* = \ell^\infty$, consequently $J(B_{\ell^1})$ is $\sigma((\ell^p)^{**}, \ell^\infty)$-dense in $B_{(\ell^p)^{**}}$ using Goldstine's theorem. But $B_{\ell^p} \subseteq B_{\ell^1}$, so the claim follows. I hope that this is correct and what you had in mind. $\endgroup$ Sep 11, 2023 at 22:41


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