Coercive function

{{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 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 spaces <math>X</math> and <math>X'</math> is called '''coercive''' if for every 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 composition of a 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 functions * Lax-Milgram lemma

==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 }}

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