Suppose we have a pseudocompact, Hausdorff space $L$ (pseudocompact means that each continuous function $f\colon L\to \mathbb{R}$ is bounded). Consider the space $C(L)$ of continuous real-valued functions on $L$. It seems that it is a Banach space (the proof for compact spaces should carry on).
Assuming $L$ is locally compact, is $C(L)$ isomorphic to $C_0(L)$, where $C_0(L)$ stands for the Banach space of continuous functions vanishing at infinity?