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

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

Title	coercive function
Canonical name	CoerciveFunction
Date of creation	2013-03-22 15:20:13
Last modified on	2013-03-22 15:20:13
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	5
Author	paolini (1187)
Entry type	Definition
Classification	msc 54A05
Synonym	coercive
Synonym	coercitive
Synonym	coercitive function