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
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(x) = (-x2, x1) by 90° is a norm-coercive mapping which fails to be a coercive vector field since for every .
Coercive operators and forms
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.
A mapping between two normed vector spaces and is called norm-coercive iff
(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
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