In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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'' : R
''n'' → R
''n'' is called coercive if
:
where "
" denotes the usual
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
and
denotes the usual Euclidean
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of the vector ''x''.
A coercive vector field is in particular norm-coercive since
for
, by
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
.
However a norm-coercive mapping
''f'' : R
''n'' → R
''n''
is not necessarily a coercive vector field. For instance
the rotation
''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
for every
.
Coercive operators and forms
A
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
where
is a real
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, is called coercive if there exists a constant
such that
:
for all
in
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 linear i ...
is called coercive if there exists a constant
such that
:
for all
in
It follows from the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the R ...
that any symmetric (defined as
for all
in
), continuous (
for all
in
and some constant
) and coercive bilinear form
has the representation
:
for some self-adjoint operator
which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator
the bilinear form
defined as above is coercive.
If
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,
for big
(if
is bounded, then it readily follows); then replacing
by
we get that
is a coercive operator.
One can also show that the converse holds true if
is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.
Norm-coercive mappings
A mapping
between two normed vector spaces
and
is called norm-coercive iff
:
.
More generally, a function
between two
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s
and
is called coercive if for every
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
of
there exists a compact subset
of
such that
:
The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definitio ...
followed by a coercive map is coercive.
(Extended valued) coercive functions
An (extended valued) function
is called coercive if
:
A real valued coercive function
is, in particular, norm-coercive. However, a norm-coercive function
is not necessarily coercive.
For instance, the identity function on
is norm-coercive
but not coercive.
See also:
radially unbounded function
In mathematics, a radially unbounded function is a function f: \mathbb^n \rightarrow \mathbb for which
:\, x\, \to \infty \Rightarrow f(x) \to \infty.
Or equivalently,
:\forall c > 0:\exists r > 0 : \forall x \in \mathbb^n: _r_\Rightarrow_f ...
s
References
*
*
*
{{PlanetMath attribution, id=7154, title=Coercive Function
Functional analysis
General topology
Types of functions