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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 if \frac \to + \infty \text \, x \, \to + \infty, where "\cdot" denotes the usual
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
and \, x\, denotes the usual Euclidean norm of the vector ''x''. A coercive vector field is in particular norm-coercive since \, f(x)\, \geq (f(x) \cdot x) / \, x \, for x \in \mathbb^n \setminus \ , by
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
. However a norm-coercive mapping is not necessarily a coercive vector field. For instance the rotation by 90° is a norm-coercive mapping which fails to be a coercive vector field since f(x) \cdot x = 0 for every x \in \mathbb^2.


Coercive operators and forms

A self-adjoint operator A:H\to H, where H is a real
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, is called coercive if there exists a constant c>0 such that \langle Ax, x\rangle \ge c\, x\, ^2 for all x in H. A
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
a:H\times H\to \mathbb R is called coercive if there exists a constant c>0 such that a(x, x)\ge c\, x\, ^2 for all x in H. It follows from the Riesz representation theorem that any symmetric (defined as a(x, y)=a(y, x) for all x, y in H), continuous (, a(x, y), \le k\, x\, \,\, y\, for all x, y in H and some constant k>0) and coercive bilinear form a has the representation a(x, y)=\langle Ax, y\rangle for some self-adjoint operator A:H\to H, which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator A, the bilinear form a defined as above is coercive. If A:H\to H 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, \langle Ax, x\rangle \ge C\, x\, for big \, x\, (if \, x\, is bounded, then it readily follows); then replacing x by x\, x\, ^ we get that A is a coercive operator. One can also show that the converse holds true if A is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.


Norm-coercive mappings

A mapping f : X \to X' between two normed vector spaces (X, \, \cdot \, ) and (X', \, \cdot \, ') is called norm-coercive if and only if \, f(x)\, ' \to + \infty \mbox \, x\, \to +\infty . More generally, a function f : X \to X' between two
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s X and X' is called coercive if for every compact subset K' of X' there exists a compact subset K of X such that f (X \setminus K) \subseteq X' \setminus K'. The composition of a
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
proper map In mathematics, a function (mathematics), function between topological spaces is called proper if inverse images of compact space, compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition ...
followed by a coercive map is coercive.


(Extended valued) coercive functions

An (extended valued) function f: \mathbb^n \to \mathbb \cup \ is called coercive if f(x) \to + \infty \mbox \, x \, \to + \infty. A real valued coercive function f:\mathbb^n \to \mathbb is, in particular, norm-coercive. However, a norm-coercive function f:\mathbb^n \to \mathbb is not necessarily coercive. For instance, the identity function on \mathbb is norm-coercive but not coercive.


See also

* Radially unbounded functions


References

* * * {{PlanetMath attribution, id=7154, title=Coercive Function Functional analysis General topology Types of functions