In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use.

Coercive vector fields

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A vector field f : RnRn is called coercive if   where " " denotes the usual dot product and   denotes the usual Euclidean norm of the vector x.

A coercive vector field is in particular norm-coercive since   for  , by Cauchy–Schwarz inequality. However a norm-coercive mapping f : RnRn is not necessarily a coercive vector field. For instance the rotation f : R2R2, f(x) = (−x2, x1) by 90° is a norm-coercive mapping which fails to be a coercive vector field since   for every  .

Coercive operators and forms

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A self-adjoint operator   where   is a real Hilbert space, is called coercive if there exists a constant   such that   for all   in  

A bilinear form   is called coercive if there exists a constant   such that   for all   in  

It follows from the Riesz representation theorem that any symmetric (defined as   for all   in  ), continuous (  for all   in   and some constant  ) and coercive bilinear form   has the representation  

for some self-adjoint operator   which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator   the bilinear form   defined as above is coercive.

If   is a coercive operator then it is a coercive mapping (in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product). Indeed,   for big   (if   is bounded, then it readily follows); then replacing   by   we get that   is a coercive operator. One can also show that the converse holds true if   is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

Norm-coercive mappings

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A mapping   between two normed vector spaces   and   is called norm-coercive if and only if  

More generally, a function   between two topological spaces   and   is called coercive if for every compact subset   of   there exists a compact subset   of   such that  

The composition of a bijective proper map followed by a coercive map is coercive.

(Extended valued) coercive functions

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An (extended valued) function   is called coercive if   A real valued coercive function   is, in particular, norm-coercive. However, a norm-coercive function   is not necessarily coercive. For instance, the identity function on   is norm-coercive but not coercive.

See also

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References

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  • Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations (Second ed.). New York, NY: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0.
  • Bashirov, Agamirza E (2003). Partially observable linear systems under dependent noises. Basel; Boston: Birkhäuser Verlag. ISBN 0-8176-6999-X.
  • Gilbarg, D.; Trudinger, N. (2001). Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer. ISBN 3-540-41160-7.

This article incorporates material from Coercive Function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.