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I know that the vector space of all real valued continuous functions on a compact Hausdorff space can be infinite dimensional. When will it be finite dimensional? And how will I identify that vector space with $\mathbb{R}^n$ for some $n$?

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    You probably want to clarify what you mean. Your first statement says the vector space is always infinite dimensional but then you ask when is it finite dimensional. In any case, if your space is finite then the vector space is finite dimensional, with the dimension equal to the number of points in your space.2011-03-09

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