Coercive function

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In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. More precisely, a function f : Rⁿ → Rⁿ is called coercive if

$\frac{f(u) \cdot u}{| u |} \to + \infty \mbox{ as } | u | \to + \infty,$

where " $\cdot$ " denotes the usual dot product and "| u |" denotes the usual Euclidean norm of the vector u.

More generally, a function f : X → Y between two topological spaces X and Y is called coercive if, for every compact subset J of Y there exists a compact subset K of X such that

$f (X \setminus K) \subseteq Y \setminus J.$

[edit] References

Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, New York, NY: Springer-Verlag, xiv+434. ISBN 0-387-00444-0.