coercive function

Definition 1 (coercive function).

Let $X$ and $Y$ be topological spaces.A function $f\mathrm{:}X\mathrm{\to }Y$ is said to be coercive if for every compact set $J\mathrm{\subset }Y$ there exists a compact set $K\mathrm{\subset }X$ 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 ${\mathrm{R}}^n$).

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

$$\underset{|x|\to +\mathrm{\infty }}{lim}|f(x)|=+\mathrm{\infty }.$$