Coercive function
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" : R → R is called coercive if where "formula_2" denotes the usual dot product and formula_3 denotes the usual Euclidean norm of the vector "x". A coercive vector field is in particular norm-coercive since formula_4 for formula_5, by Cauchy–Schwarz inequality. However a norm-coercive mapping "f" : R → R is not necessarily a coercive  vector field. For instance the rotation "f" : R → R, "f(x) = (-x, x)" by 90° is a norm-coercive mapping which fails to be a coercive vector field since formula_6 for every formula_7. Coercive operators and forms. A self-adjoint operator formula_8 where formula_9 is a real Hilbert space, is called coercive if there exists a constant formula_10 such that for all formula_12 in formula_13 A bilinear form formula_14 is called coercive if there exists a constant formula_10 such that for all formula_12 in formula_13 It follows from the Riesz representation theorem that any symmetric (defined as:formula_19  for all formula_20 in formula_9), continuous (formula_22 for all formula_20 in formula_9 and some constant formula_25) and coercive bilinear form formula_26 has the representation for some self-adjoint operator formula_8 which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator formula_29 the bilinear form formula_26 defined as above is coercive. If formula_31 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, formula_32 for big formula_3 (if formula_3 is bounded , then it readily follows); then replacing formula_12 by formula_36 we get that formula_37 is a coercive operator. One can also show that the converse holds true if formula_37 is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible. Norm-coercive mappings. A mapping formula_39 between two normed vector spaces formula_40 and formula_41 is called norm-coercive iff More generally, a function formula_39 between two topological spaces formula_44 and formula_45 is called coercive if for every compact subset formula_46 of formula_45 there exists a compact subset formula_48 of formula_44 such that The  composition of a bijective proper map followed by a coercive map is coercive. Section::::(Extended valued) coercive functions. An (extended valued) function formula_51 is called coercive if A real valued coercive function formula_53 is, in particular, norm-coercive. However, a norm-coercive function formula_53 is not necessarily coercive. For instance, the identity function on formula_55 is norm-coercive but not coercive. See also: radially unbounded functions