# Vanish at infinity

In mathematics, a function on a normed vector space is said to **vanish at infinity** if

- as

For example, the function

defined on the real line vanishes at infinity.

More generally, a function on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number , there is a compact subset such that

whenever the point lies outside of .

In the other words,for each positive number the set is compact.

For a given locally compact space , the set of such functions

(where is either the field of real numbers or the field of complex numbers) forms an -vector space with respect to pointwise scalar multiplication and addition, often denoted .

Both of these notions correspond to the intuitive notion of adding a point at infinity and requiring the values of the function to get arbitrarily close to zero as we approach it. This definition can be formalized in many cases by adding a point at infinity.

## Rapidly decreasing

Refining the concept, one can look more closely to the *rate of vanishing* of functions at infinity. One of the basic intuitions of mathematical analysis is that the Fourier transform interchanges smoothness conditions with rate conditions on vanishing at infinity. The **rapidly decreasing** test functions of tempered distribution theory are smooth functions that are

- o(|
*x*|^{−N})

for all *N*, as |*x*| → ∞, and such that all their partial derivatives satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding distribution theory of *tempered distributions* will have the same good property.

## References

This article needs additional citations for verification. (January 2008) |

- Hewitt, E and Stromberg, K (1963).
*Real and abstract analysis*. Springer-Verlag.