---
tag: N59BW1QFD1
kind: concept
name: Space of functions vanishing at infinity
msc: ["46E15", 54C35, 46B25]
specializes: [MG3X8Y1VT4]
sources:
  - url: https://en.wikipedia.org/wiki/Vanish_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|MG3X8Y1VT4]], 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.

## Explanations

### L1 human

A function vanishes at infinity when it gets as small as you like once you go far enough away.
On $\R$, the bell curve $e^{-x^2}$ vanishes at infinity, and so does $1/(1 + x^2)$.
The constant function $1$ does not.
On the set $\N$ of natural numbers, these functions are just the sequences that tend to $0$.