54A05-CoerciveFunction.tex

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\pmcanonicalname{CoerciveFunction}
\pmcreated{2013-03-22 15:20:13}
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\pmtitle{coercive function}
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\pmtype{Definition}
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\pmsynonym{coercive}{CoerciveFunction}
\pmsynonym{coercitive}{CoerciveFunction}
\pmsynonym{coercitive function}{CoerciveFunction}

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\begin{document}
\begin{definition}[coercive function]
Let $X$ and $Y$ be topological spaces.
A function $f\colon X\to Y$ is said to be \emph{coercive} if for every compact set $J\subset Y$ there exists a compact set $K\subset X$ such that
\[
  F(X\setminus K) \subset Y\setminus J.
\]
\end{definition}

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

\begin{proposition}[coercive functions on $\R^n$]
A function $f\colon \R^n \to \R^m$ is coercive if and only if 
\[
  \lim_{|x|\to +\infty} |f(x)| = +\infty.
\]
\end{proposition}
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\end{document}