Let $K$ be a non-separable compact topological space. Is there a subspace of $C(K)$ which is isomorphic to $\ell^{\infty}$? (Where $C(K)$ is the Banach space of continuous functions on $K$.)
2026-04-04 15:32:47.1775316767
Space of continuous functions on compact non-separable spaces and $\ell^{\infty}$
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No, this need not be the case. Take $K$ to be any weakly compact subset of a Banach space. Then the space $C(K)$ is weakly compactly generated. However $\ell_\infty$ does not embed into a weakly compactly generated Banach space.
As for specific examples, you may take unit ball of non-separable reflexve spaces with weak topology.