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# 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

$\lim_{{|x|\to+\infty}}|f(x)|=+\infty.$ |

Synonym:

coercive, coercitive, coercitive function

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

54A05*no label found*

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