# coercive function

###### Definition 1 (coercive function).

Let $X$ and $Y$ be topological spaces. A function $f\colon X\to Y$ is said to be coercive if for every compact set $J\subset Y$ there exists a compact set $K\subset X$ such that

 $F(X\setminus K)\subset Y\setminus J.$

The general definition given above has a clear sense when specialized to the Euclidean spaces, as shown in the following result.

###### Proposition 1 (coercive functions on $\mathbb{R}^{n}$).

A function $f\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is coercive if and only if

 $\lim_{|x|\to+\infty}|f(x)|=+\infty.$
Title coercive function CoerciveFunction 2013-03-22 15:20:13 2013-03-22 15:20:13 paolini (1187) paolini (1187) 5 paolini (1187) Definition msc 54A05 coercive coercitive coercitive function