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If $X$ is compact Hausdorff space and $C(X)$ is the set of all continuous complex valued functions on $X$,then prove that $C(X)$ is finite dimensional if and only if $X$ is finite.

My problem:If we suppose $X=\{x_1,x_2,...,x_n\}$ then how we can prove that $C(X)$ is spanned by the functions $e_i(x_j)=\delta_{ij}$ and $C(X)\cong \mathbb{C^n}$? And if we suppose $X$ is infinite then can I use the solution (When is the vector space of continuous functions on a compact Hausdorff space finite dimensional?) by replacing $\mathbb{R}$ by $\mathbb{C}$?

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    Once you observe that the Tietze extension theorem can be applied to a complex-valued function $f$ by applying it separately to the real part $\Re f$ and the imaginary part $\Im f$, which are real-valued, you can then safely go through Pete L. Clark's linked-to answer, which answers both parts of your question, and replace $\mathbb{R}$ by $\mathbb{C}$. – Branimir Ćaćić Aug 25 '14 at 11:19
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    As for why $C({x_1,\dotsc,x_n})$ is spanned by ${e_i}$, it's basically the same argument as for why the dual basis to a basis of a finite-dimensional vector space $V$ actually yields a spanning set for the dual space $V^\ast$. In short, for any $f \in C({x_1,\dotsc,x_n})$, just check that $ f = \sum_{j=1}^n f(x_j) e_j. $ – Branimir Ćaćić Aug 25 '14 at 11:22
  • Yes, you can just replace $\mathbb R$ by $\mathbb C$. (You don't really need to worry about real and imaginary parts.) Hence this is essentially a duplicate. – Jonas Meyer Aug 26 '14 at 02:43

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