Let K be a compact Hausdorff space with weight w(K), τ an infinite cardinal with cofinality cf(τ) and X a Banach space. In contrast with a classical theorem of Cembranos and Freniche it is shown that if cf$(\tau )>$ w(K) then the space $C(K,X)$ contains a complemented copy of $c_0(\tau )$ if and only if X does.

This result is optimal for every infinite cardinal τ, in the sense that it can not be improved by replacing the inequality cf$(\tau )>$ w(K) by another weaker than it.

中文翻译：

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何时Ç（ķ，X）含有的可补拷贝Ç ₀（Γ）当且仅当X呢？

令K为权重为w（K）的紧凑Hausdorff空间，τ为协定性cf（τ）的无穷基数，X为Banach空间。与Cembranos和Freniche的经典定理相比，如果$（\tau ）>$w（K）然后空间$C（ķ，X）$ 包含的补充副本 $C_0（\tau ）$当且仅当X这样做时。

对于每个无限大基数τ而言，此结果都是最佳的，因为无法通过替换不等式cf来改善该结果$（\tau ）>$w（K）比其弱。