Function vanishing at infinity

From Encyclopedia of Mathematics
Jump to: navigation, search

Let $X$ be a topological space. A real- or complex-valued function on $X$ is said to vanish at infinity if for each $\epsilon>0$ there is a compact set $K_\epsilon$ such that $|f(x)|<\epsilon$ for all $x \in X \setminus K_\epsilon$. For non-compact $X$, such a function can be extended to a continuous function on the one-point compactification $X^* = X \cup \{\star\}$ of $X$ (with value $0$ at $\star$).

The algebra of functions on $X$ vanishing at infinity is denoted by $C_0(X)$. In many cases $C_0(X)$ determines $X$, see e.g. Banach–Stone theorem. If $X$ is compact, $C_0(X) = C(X)$. The space $C_0(X)$ identifies with $\{f \in X^* : f(\star)=0 \}$.


[a1] E. Behrends, "M-structure and the Banach–Stone theorem" , Springer (1979)
[a2] K. Jarosz, "Perturbations of Banach spaces" , Springer (1985)
How to Cite This Entry:
Function vanishing at infinity. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article