Difference between revisions of "Talk:Function vanishing at infinity"
From Encyclopedia of Mathematics
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"In many cases $C_0(X)$ determines $X$, see e.g. Banach–Stone theorem" — Yes, but in Banach-Stone theorem $C_0(X)$ is treated as a Banach space (rather than Banach algebra; in the latter case the claim is almost trivial). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:37, 10 January 2018 (CET) | "In many cases $C_0(X)$ determines $X$, see e.g. Banach–Stone theorem" — Yes, but in Banach-Stone theorem $C_0(X)$ is treated as a Banach space (rather than Banach algebra; in the latter case the claim is almost trivial). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:37, 10 January 2018 (CET) | ||
:The concept of function vanishing at infinity is mentioned in the Banach–Stone theorem article but this does not seem particularly important point to make here. I suppose it's about reconstructing the space from the maximal ideals of the algebra, and then the functions that vanish at infinity form an ideal which does not correspond to a point of the space. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 08:38, 11 January 2018 (CET) | :The concept of function vanishing at infinity is mentioned in the Banach–Stone theorem article but this does not seem particularly important point to make here. I suppose it's about reconstructing the space from the maximal ideals of the algebra, and then the functions that vanish at infinity form an ideal which does not correspond to a point of the space. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 08:38, 11 January 2018 (CET) | ||
| + | ::Yes, given the algebra, you can easily reconstruct the space from the maximal ideals. It is not so easy to reconstruct the space when multiplication is forgotten, only linear structure and norm are given; this is Banach-Stone theorem. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 09:57, 11 January 2018 (CET) | ||
Latest revision as of 08:57, 11 January 2018
"In many cases $C_0(X)$ determines $X$, see e.g. Banach–Stone theorem" — Yes, but in Banach-Stone theorem $C_0(X)$ is treated as a Banach space (rather than Banach algebra; in the latter case the claim is almost trivial). Boris Tsirelson (talk) 22:37, 10 January 2018 (CET)
- The concept of function vanishing at infinity is mentioned in the Banach–Stone theorem article but this does not seem particularly important point to make here. I suppose it's about reconstructing the space from the maximal ideals of the algebra, and then the functions that vanish at infinity form an ideal which does not correspond to a point of the space. Richard Pinch (talk) 08:38, 11 January 2018 (CET)
- Yes, given the algebra, you can easily reconstruct the space from the maximal ideals. It is not so easy to reconstruct the space when multiplication is forgotten, only linear structure and norm are given; this is Banach-Stone theorem. Boris Tsirelson (talk) 09:57, 11 January 2018 (CET)
How to Cite This Entry:
Function vanishing at infinity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_vanishing_at_infinity&oldid=42711
Function vanishing at infinity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_vanishing_at_infinity&oldid=42711