# Coercive function

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

A vector field *f* : **R**^{n} → **R**^{n} 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* : **R**^{n} → **R**^{n}
is not necessarily a coercive vector field. For instance
the rotation
*f* : **R**^{2} → **R**^{2}, *f(x) = (-x _{2}, x_{1})*
by 90° is a norm-coercive mapping which fails to be a coercive vector field since
for every .

## Coercive operators and forms

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

A mapping
between two normed vector spaces
and
is called **norm-coercive** iff

- .

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

An (extended valued) function
is called **coercive** iff

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: radially unbounded functions

## References

- 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.*