Vanish at infinity

In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from this difference, 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 one approaches it. This definition can be formalized in many cases by adding an (actual) point at infinity.

A function on a normed vector space is said to vanish at infinity if the function approaches ${\displaystyle 0}$ as the input grows without bounds (that is, ${\displaystyle f(x)\to 0}$ as ${\displaystyle \|x\|\to \infty }$). Or,

in the specific case of functions on the real line.

For example, the function

defined on the real line vanishes at infinity.

Alternatively, a function ${\displaystyle f}$ on a locally compact space ${\displaystyle \Omega }$ vanishes at infinity, if given any positive number ${\displaystyle \varepsilon >0}$, there exists a compact subset ${\displaystyle K\subseteq \Omega }$ such that

whenever the point ${\displaystyle x}$ lies outside of ${\displaystyle K.}$ In other words, for each positive number ${\displaystyle \varepsilon >0}$, the set ${\displaystyle \left\{x\in \Omega :\|f(x)\|\geq \varepsilon \right\}}$ has compact closure. For a given locally compact space ${\displaystyle \Omega }$ the set of such functions

valued in ${\displaystyle \mathbb {K} ,}$ which is either ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} ,}$ forms a ${\displaystyle \mathbb {K} }$-vector space with respect to pointwise scalar multiplication and addition, which is often denoted ${\displaystyle C_{0}(\Omega ).}$