View source for Function vanishing at infinity

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 $\{f \in X^* : f(\star)=0 \}$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Behrends,   "M-structure and the Banach–Stone theorem" , Springer  (1979)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jarosz,   "Perturbations of Banach spaces" , Springer  (1985)</TD></TR>
</table>

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