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 : Rn → Rn 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 : Rn → Rn
is not necessarily a coercive vector field. For instance
the rotation
f : R2 → R2, f =
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, continuous 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. Indeed, for big ; 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 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