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.
A vector field f : Rn → Rn is called coercive if
" 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 the rotation f : R2 → R2, f(x) = (−x2, x1) by 90° is a norm-coercive mapping which fails to be a coercive vector field since
A self-adjoint operator
is a real Hilbert space, is called coercive if there exists a constant
is called coercive if there exists a constant
It follows from the Riesz representation theorem that any symmetric (defined as
) and coercive bilinear form
which then turns out to be a coercive operator.
Also, given a coercive self-adjoint operator
the bilinear form
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).
is bounded, then it readily follows); then replacing
One can also show that the converse holds true if
The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.
between two normed vector spaces
between two topological spaces
is called coercive if for every compact subset
there exists a compact subset
The composition of a bijective proper map followed by a coercive map is coercive.
An (extended valued) function
A real valued coercive function
However, a norm-coercive function
For instance, the identity function on
This article incorporates material from Coercive Function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.