# 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

${\displaystyle {\frac {f(x)\cdot x}{\|x\|}}\to +\infty {\mbox{ as }}\|x\|\to +\infty ,}$

where "${\displaystyle \cdot }$" denotes the usual dot product and ${\displaystyle \|x\|}$ denotes the usual Euclidean norm of the vector x.

A coercive vector field is in particular norm-coercive since ${\displaystyle \|f(x)\|\geq (f(x)\cdot x)/\|x\|}$ for ${\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}}$, 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 ${\displaystyle f(x)\cdot x=0}$ for every ${\displaystyle x\in \mathbb {R} ^{2}}$.

## Coercive operators and forms

A self-adjoint operator ${\displaystyle A:H\to H,}$ where ${\displaystyle H}$ is a real Hilbert space, is called coercive if there exists a constant ${\displaystyle c>0}$ such that

${\displaystyle \langle Ax,x\rangle \geq c\|x\|^{2}}$

for all ${\displaystyle x}$ in ${\displaystyle H.}$

A bilinear form ${\displaystyle a:H\times H\to \mathbb {R} }$ is called coercive if there exists a constant ${\displaystyle c>0}$ such that

${\displaystyle a(x,x)\geq c\|x\|^{2}}$

for all ${\displaystyle x}$ in ${\displaystyle H.}$

It follows from the Riesz representation theorem that any symmetric (defined as:${\displaystyle a(x,y)=a(y,x)}$ for all ${\displaystyle x,y}$ in ${\displaystyle H}$), continuous (${\displaystyle |a(x,y)|\leq k\|x\|\,\|y\|}$ for all ${\displaystyle x,y}$ in ${\displaystyle H}$ and some constant ${\displaystyle k>0}$) and coercive bilinear form ${\displaystyle a}$ has the representation

${\displaystyle a(x,y)=\langle Ax,y\rangle }$

for some self-adjoint operator ${\displaystyle A:H\to H,}$ which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator ${\displaystyle A,}$ the bilinear form ${\displaystyle a}$ defined as above is coercive.

If ${\displaystyle A:H\to H}$ 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, ${\displaystyle \langle Ax,x\rangle \geq C\|x\|}$ for big ${\displaystyle \|x\|}$ (if ${\displaystyle \|x\|}$ is bounded, then it readily follows); then replacing ${\displaystyle x}$ by ${\displaystyle x\|x\|^{-2}}$ we get that ${\displaystyle A}$ is a coercive operator. One can also show that the converse holds true if ${\displaystyle A}$ is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

## Norm-coercive mappings

A mapping ${\displaystyle f:X\to X'}$ between two normed vector spaces ${\displaystyle (X,\|\cdot \|)}$ and ${\displaystyle (X',\|\cdot \|')}$ is called norm-coercive iff

${\displaystyle \|f(x)\|'\to +\infty {\mbox{ as }}\|x\|\to +\infty }$.

More generally, a function ${\displaystyle f:X\to X'}$ between two topological spaces ${\displaystyle X}$ and ${\displaystyle X'}$ is called coercive if for every compact subset ${\displaystyle K'}$ of ${\displaystyle X'}$ there exists a compact subset ${\displaystyle K}$ of ${\displaystyle X}$ such that

${\displaystyle f(X\setminus K)\subseteq X'\setminus K'.}$

The composition of a bijective proper map followed by a coercive map is coercive.

## (Extended valued) coercive functions

An (extended valued) function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} \cup \{-\infty ,+\infty \}}$ is called coercive if

${\displaystyle f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .}$

A real valued coercive function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is, in particular, norm-coercive. However, a norm-coercive function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is not necessarily coercive. For instance, the identity function on ${\displaystyle \mathbb {R} }$ is norm-coercive but not coercive.