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 is called coercive" denotes the usual dot product denotes the usual Euclidean norm of the vector x.

A coercive vector field, by Cauchy–Schwarz inequality. However a norm-coercive mapping is not necessarily a coercive vector field. For instance the rotation .

Coercive operators and forms

A self-adjoint operator is a real Hilbert space, is called coercive

A bilinear form is called coercive

It follows from the Riesz representation theorem

defined as above is coercive.

is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

Norm-coercive mappings

is called norm-coercive

between two topological spaces is called coercive if for every compact subset

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

(Extended valued) coercive functions

is called coercive is not necessarily coercive. For instance, the identity function is norm-coercive but not coercive.

See also

• Radially unbounded functions

• Lax-Milgram lemma

References