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I've recently read about the notion of (Rademacher) type and cotype of a Banach space in some article. In the books I checked afterwards, typical examples studied were $L^p$-spaces or the Schatten classes but nothing was said about spaces of continuous functions. As these are arguably one of the most important examples of Banach spaces, I wonder why this is.

So here is my question: Is anything known about the type and/or cotype of the Banach space $C(K)$ with suitable $K$?

I'm particularly interested in the case where $K$ is a compact interval on the real line.

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    $\begingroup$ It is known that $C(K)$, for infinite $K$, contains a copy of $c_0$, hence it does not have nontrivial type (meaning $>1$) or nontrivial cotype (meaning $<\infty$). $\endgroup$
    – Dirk Werner
    Commented Feb 13, 2020 at 13:03
  • $\begingroup$ @DirkWerner Thank you very much for this comment, so this means I could not find it in the books as the answer is trivial (to somebody who knows more functional analysis than I do...). If you want you could extend your comment to an answer and I will accept it. $\endgroup$
    – F. Carbon
    Commented Feb 13, 2020 at 14:28
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    $\begingroup$ This is just an addendum to Dirk Werner’s definitive answer but it might add useful information: every Banach space is isometrically isomorphic to a subspace of a $C(K)$-space (even $C([0,1])$ if separable) so there is no point in examining special properties of the latter if they are inherited by subspaces. $\endgroup$
    – user131781
    Commented Feb 13, 2020 at 15:45
  • $\begingroup$ @user131781 this is a very useful comment, thank you very much! $\endgroup$
    – F. Carbon
    Commented Feb 13, 2020 at 16:06

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It is known that $C(K)$, for infinite $K$, contains a copy of $c_0$, hence it does not have nontrivial type (meaning $>1$) or nontrivial cotype (meaning $<\infty$).

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