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Does anyone know where I can find a reference for the following well-known fact:

Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s.


$c_0(I, C(X_i)) = C_0(X)$,

where the left hand side stands for the $c_0$-sum (possibly uncountable) of $C(X_i)$s.

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It might be difficult to locate this formally proven in the literature. I only saw this stated as an exercise or as an obvious (or well-known) fact. For example, Conway's Course in Functional Analysis, chapter VII.1, exercise 9. – Martin Jan 30 '13 at 21:44
I could provide a proof, if you want, but you seem to be more interested in a reference. – Martin Jan 30 '13 at 21:49
Thanks! The proof is rather easy. I think this is enough for me, you can post is as an answer. – user60253 Jan 30 '13 at 22:04
up vote 2 down vote accepted

Since the result is quite easy to prove, it is mostly stated as an exercise or as an obvious fact.

For example, Conway, A Course in Functional Analysis, Chapter VII.1, Exercise 9., page 192 states the slightly more general:

Let $\{X_i : i \in I\}$ be a collection of locally compact spaces and let $X = $ the disjoint union of these spaces furnished with the topology $\{U \subseteq X : U \cap X_i \text{ is open in } X_i \text{ for all } i\}$. Show that $X$ is locally compact and $\bigoplus_0 C_0(X_i)$ is isometrically isomorphic to $C_0(X)$.

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