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.
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
However a norm-coercive mapping
f : Rn → Rn
is not necessarily a coercive vector field. For instance
f : R2 → R2, f(x) = (-x2, x1)
by 90° is a norm-coercive mapping which fails to be a coercive vector field since
for every .
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.
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.