# 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 : RnRn is called coercive if

${\frac {f(x)\cdot x}{\|x\|}}\to +\infty {\mbox{ as }}\|x\|\to +\infty ,$ A coercive vector field is in particular norm-coercive since $\|f(x)\|\geq (f(x)\cdot x)/\|x\|$ for $x\in {\mathbb {R} }^{n}\setminus \{0\}$ , by Cauchy Schwarz inequality. However a norm-coercive mapping f : RnRn is not necessarily a coercive vector field. For instance the rotation f : R2R2, f(x) = (-x2, x1) by 90° is a norm-coercive mapping which fails to be a coercive vector field since $f(x)\cdot x=0$ for every $x\in {\mathbb {R} }^{2}$ .

## Coercive operators and forms

$\langle Ax,x\rangle \geq c\|x\|^{2}$ $a(x,x)\geq c\|x\|^{2}$ $a(x,y)=\langle Ax,y\rangle$ 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.

One can also show that any self-adjoint operator $A:H\to H$ 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.

## Norm-coercive mappings

$\|f(x)\|'\to +\infty {\mbox{ as }}\|x\|\to +\infty$ .
$f(X\setminus K)\subseteq X'\setminus K'.$ The composition of a bijective proper map followed by a coercive map is coercive.

## (Extended valued) coercive functions

$f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .$ A realvalued coercive function $f:{\mathbb {R} }^{n}\to {\mathbb {R} }$ is in particular norm-coercive. However a norm-coercive function $f:{\mathbb {R} }^{n}\to {\mathbb {R} }$ is not necessarily coercive. For instance the identity function on ${\mathbb {R} }$ is norm-coercive but not coercive.