A map between spaces (say, a continuous map between topological spaces) vanishes at infinity if gets arbitrarily close to zero as gets sufficiently close to infinity.
For a map , we need a notion of being close to in , so take to be a pointed space; then getting arbitrarily close to means entering any neighbourhood of the basepoint. We also need a notion of being close to infinity in , so take to be a locally compact Hausdorff space; then getting sufficiently close to infinity means entering the exterior of some compact subspace. (To interpret ‘getting’, of course, we may use nets.) It is likely, however, that further generalisations are possible.
Let and be topological spaces, and let be a continuous map (or potentially any function) from to . Let be pointed, and let be locally compact Hausdorff.
The map vanishes at infinity if for every neighbourhood of the basepoint in , there is compact subspace of such that belongs to whenever lies in the exterior of in .
In case is a pointed metric space (such as a Banach space, with basepoint ; or in particular the real line, with basepoint ), then we may equivalently say:
(Here, is the norm in a Banach space, or more generally the distance from the basepoint in any pointed metric space.)
One way of considering this definition is that one can adjoin to a point “at infinity”, denoted , by declaring that the open neighborhoods of are sets of the form for compact. This is called the one-point compactification, denoted . Then a continuous function vanishes at infinity equivalently if it extends to a map , continuous at (at least) that sends to – thus literally- “vanishing at ”.
For more see
at one-point compactification – Relevance for Monopoles and Instantons
at Yang-Mills instanton – SU(2)-instantons from the correct maths to the traditional physics story.
Last revised on October 21, 2019 at 06:00:47. See the history of this page for a list of all contributions to it.