imported>MichaelMaggs: Adding local short description: "Mathematical function", overriding Wikidata description "mathematical function that "grows rapidly" at the extremes of the space on which it is defined"

2024-11-21T19:21:37Z

Adding local short description: "Mathematical function", overriding Wikidata description "mathematical function that "grows rapidly" at the extremes of the space on which it is defined"

New page

{{Short description|Mathematical function}}
{{no footnotes|date=April 2023}}
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 {{math|''f'' : '''R'''''n'' → '''R'''''n''}} is called '''coercive''' if
<math display="block">\frac{f(x) \cdot x}{\| x \|} \to + \infty \text{ as } \| x \| \to + \infty,</math>
where "<math>\cdot</math>" denotes the usual [[dot product]] and <math>\|x\|</math> denotes the usual Euclidean [[norm (mathematics)|norm]] of the vector ''x''.

A coercive vector field is in particular norm-coercive since <math>\|f(x)\| \geq (f(x) \cdot x) / \| x \|</math> for <math>x \in \mathbb{R}^n \setminus \{0\} </math>, by [[Cauchy–Schwarz inequality]].
However a norm-coercive mapping {{math|''f'' : '''R'''''n'' → '''R'''''n''}} is not necessarily a coercive vector field. For instance the rotation {{math|1=''f'' : '''R'''''2'' → '''R'''''2'', ''f''(''x'') = (−''x''2, ''x''1)}} by 90° is a norm-coercive mapping which fails to be a coercive vector field since <math>f(x) \cdot x = 0</math> for every <math>x \in \mathbb{R}^2</math>.

==Coercive operators and forms==
A [[self-adjoint operator]] <math>A:H\to H,</math> where <math>H</math> is a real [[Hilbert space]], is called '''coercive''' if there exists a constant <math>c>0</math> such that
<math display="block">\langle Ax, x\rangle \ge c\|x\|^2</math>
for all <math>x</math> in <math>H.</math>

A [[bilinear form]] <math>a:H\times H\to \mathbb R</math> is called '''coercive''' if there exists a constant <math>c>0</math> such that
<math display="block">a(x, x)\ge c\|x\|^2</math>
for all <math>x</math> in <math>H.</math>

It follows from the [[Riesz representation theorem]] that any symmetric (defined as <math>a(x, y)=a(y, x)</math> for all <math>x, y</math> in <math>H</math>), continuous (<math>|a(x, y)|\le k\|x\|\,\|y\|</math> for all <math>x, y</math> in <math>H</math> and some constant <math>k>0</math>) and coercive bilinear form <math>a</math> has the representation
<math display="block">a(x, y)=\langle Ax, y\rangle</math>

for some self-adjoint operator <math>A:H\to H,</math> which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator <math>A,</math> the bilinear form <math>a</math> defined as above is coercive.

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

==Norm-coercive mappings==
A mapping <math>f : X \to X' </math> between two normed vector spaces <math>(X, \| \cdot \|)</math> and <math>(X', \| \cdot \|')</math> is called '''norm-coercive''' if and only if
<math display="block"> \|f(x)\|' \to + \infty \mbox{ as } \|x\| \to +\infty .</math>

More generally, a function <math>f : X \to X' </math> between two [[topological space]]s <math>X</math> and <math>X'</math> is called '''coercive''' if for every [[compact space|compact subset]] <math>K'</math> of <math>X'</math> there exists a compact subset <math>K</math> of <math>X</math> such that
<math display="block">f (X \setminus K) \subseteq X' \setminus K'.</math>

The [[function composition|composition]] of a [[bijection|bijective]] [[proper map]] followed by a coercive map is coercive.

==(Extended valued) coercive functions==
An (extended valued) function <math>f: \mathbb{R}^n \to \mathbb{R} \cup \{- \infty, + \infty\}</math> is called '''coercive''' if
<math display="block"> f(x) \to + \infty \mbox{ as } \| x \| \to + \infty.</math>
A real valued coercive function <math>f:\mathbb{R}^n \to \mathbb{R} </math> is, in particular, norm-coercive. However, a norm-coercive function <math>f:\mathbb{R}^n \to \mathbb{R} </math> is not necessarily coercive.
For instance, the identity function on <math> \mathbb{R} </math> is norm-coercive but not coercive.

== See also ==
* [[Radially unbounded function]]s

==References==
* {{cite book|author1=Renardy, Michael |author2=Rogers, Robert C. | title=An introduction to partial differential equations | edition=Second | publisher=Springer-Verlag | location=New York, NY | year=2004 | pages=xiv+434 | isbn=0-387-00444-0 }}
* {{cite book
| last = Bashirov
| first = Agamirza E
| title = Partially observable linear systems under dependent noises
| publisher = Basel; Boston: Birkhäuser Verlag
| year = 2003
| pages =
| isbn = 0-8176-6999-X
}}
*{{cite book
| author2-link=Neil Trudinger
| first1=D.
| last1=Gilbarg
| first2=N.
| last2=Trudinger
| title = Elliptic partial differential equations of second order, 2nd ed
| publisher = Berlin; New York: Springer
| year = 2001
| pages =
| isbn = 3-540-41160-7
}}

{{PlanetMath attribution|id=7154|title=Coercive Function}}

[[Category:Functional analysis]]
[[Category:General topology]]
[[Category:Types of functions]]

Coercive function - Revision history

imported>MichaelMaggs: Adding local short description: "Mathematical function", overriding Wikidata description "mathematical function that "grows rapidly" at the extremes of the space on which it is defined"