Coercivité

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 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 self-adjoint operator where is a real Hilbert space, is called coercive if there exists a constant such that 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 if and only if More generally, a function between two topological spaces and is called coercive if for every compact subset of there exists a compact subset of such that The composition of a bijective proper map followed by a coercive map is coercive.