Let $X$ be a compact Hausdorff space. Then show that $\mathcal{C}(X)$ is reflexive iff $X$ is finite, where $\mathcal{C}(X)$ is the set of all continuous from $X$ to the base field ($\mathbb{R}$ or $\mathbb{C}$ in this case).
Attempt: If $X$ is finite, then $\mathcal{C}(X)$ is clearly finite dimensional, and hence reflexive. How do I show the converse? (Some answers seem to require Urysohn's Lemma, anyway around this?)