Difference between revisions of "Function vanishing at infinity"
From Encyclopedia of Mathematics
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| − | Let | + | 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 | + | 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==== | ====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> | + | <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> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 20:25, 10 January 2018
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
| [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: http://encyclopediaofmath.org/index.php?title=Function_vanishing_at_infinity&oldid=42708
Function vanishing at infinity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_vanishing_at_infinity&oldid=42708
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article