Problem 1. Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$.

Since $c_0$ has separable dual, such $K$ must me countable. So, we can make Problem 1 more precise:

Problem 2. Is it true that for every compact countable space $K$ the Banach space $C(K)$ is isomorphic to a subspace of $c_0$?

Another possible option:

Problem 3. Let $K$ be a compact Hausdorff space. Is it true that the Banach space $C(K)$ is isomorphic to $c_0$ if $C(K)$ is isomorphic to a subspace of $c_0$?

  • $\begingroup$ Problem 3: $K$ finite makes trivial counterexamples. Surprisingly (?) the answer is yes for $K$ infinite according to Tomek Kania's answer. $\endgroup$
    – YCor
    Jun 3, 2019 at 6:32

1 Answer 1


The Szlenk index is the answer.

A space $C(K)$, where $K$ is infinite compact Hausdorff space, is embeddable into $c_0$ if and only if $K$ is homeomorphic to an ordinal below $\omega^\omega$ and if this is the case (and $K$ is infinite) the space itself is isomorphic to $c_0$.

So the answer to problem 2 is no however the answer to problem 3 is yes.

For details see Rosenthal's chapter in the Handbook of Banach spaces.

  • 1
    $\begingroup$ Thank you very much for the answer. This is exactly what I need! $\endgroup$ Jun 3, 2019 at 5:27
  • 4
    $\begingroup$ Rosenthal, Haskell P. The Banach spaces C(K). Handbook of the geometry of Banach spaces, Vol. 2, 1547-1602, North-Holland, Amsterdam, 2003. $\endgroup$
    – YCor
    Jun 3, 2019 at 6:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.