In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
and
denotes the usual Euclidean
norm 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 an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
.
However a norm-coercive mapping is not necessarily a coercive vector field. For instance the rotation 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 where
is a real
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, 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 linea ...
is called coercive if there exists a constant
such that
for all
in
It follows from the
Riesz representation theorem 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 if and only if
More generally, a function
between two
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s
and
is called coercive if for every
compact subset of
there exists a compact subset
of
such that
The
composition of a
bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
proper map
In mathematics, a function (mathematics), function between topological spaces is called proper if inverse images of compact space, compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
...
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 functions
References
*
*
*
{{PlanetMath attribution, id=7154, title=Coercive Function
Functional analysis
General topology
Types of functions