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 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 .
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.
One can also show that any self-adjoint operator is a coercive operator if and only if 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). The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.