We prove a more general vector-valued variant of the following statement: If \(X_1\) and \(X_2\) are Choquet simplices such that the spaces of affine continuous functions \({\mathfrak {A}}(X_1, {\mathbb {R}})\) and \({\mathfrak {A}}(X_2, {\mathbb {R}})\) are isomorphic, then the cardinality of \({\text {ext}}X_1\) is equal to the cardinality of \({\text {ext}}X_2\).

中文翻译：

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Choquet 单纯形的弱向量值 Banach-Stone 定理

我们证明了以下语句的更一般的向量值变体：如果\(X_1\)和\(X_2\)是 Choquet 单纯形，使得仿射连续函数的空间\({\mathfrak {A}}(X_1, { \mathbb {R}})\)和\({\mathfrak {A}}(X_2, {\mathbb {R}})\)是同构的，那么\({\text {ext}}X_1\ )等于\({\text {ext}}X_2\)的基数。