How to prove that $(c_0(2^\kappa),\|\cdot\|_\infty)$ does not embed into $(\ell_\infty(\kappa),\|\cdot\|_\infty)$? Recall that $(c_0(2^\kappa),\|\cdot\|_\infty)$ is the Banach space of all families $(a_\lambda)_{\lambda<2^k}$ such that $\{\lambda<2^\kappa\,\colon\,|a_\lambda|\geq\varepsilon\}$ is finite for all $\varepsilon>0$.
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The space $\ell_\infty(\mathcal k)$ is a dual space, and a result by Rosenthal implies that if $X^*$ contains $c_0(\Gamma)$ then it contains $\ell_\infty(\Gamma)$, and the density character of $\ell_\infty(\Gamma)$ is strictly bigger than that of $c_0(\Gamma)$.
See Corollary 1.5 in the paper H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. (Correction, ibid., 311-313.)
answered Nov 30, 2023 at 16:15
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1$\begingroup$ Every weakly compact subset of $l_{\infty}(\kappa)$ has topological weight at most $\kappa$. The reason is that if $W$ is weakly compact then $ (W, w)$ and $(W,w^*)$ are homeomorphic and the topological weight in the $w^*$ topology of the norm bounded subsets of $l_{\infty}(\kappa)$ is at most $\kappa$.This actually holds for every dual Banach space. Now notice that the basis of $c_0 ( \Gamma ) $ together with 0 is weakly compact with topological weight equal to the cardinality of $\Gamma$. $\endgroup$ Commented Nov 30, 2023 at 22:14