Space of functions vanishing at infinity

Summary

For a locally compact Hausdorff space $X$, the space $C_0(X)$ consists of the continuous functions that become small outside compact sets. It is a closed subspace of the space of bounded continuous functions, so it is a Banach space with the supremum norm. By the Riesz–Markov–Kakutani theorem, its dual is the space of finite signed regular Borel measures on $X$, or complex ones for complex scalars. It is also the basic example of a commutative C*-algebra without unit.