Coercive function
From Wikipedia, the free encyclopedia
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. More precisely, a function f : Rn → Rn is called coercive if
where "
" denotes the usual dot product and 
denotes the usual Euclidean norm of the vector x.
More generally, a function f : X → Y between two topological spaces X and Y is called coercive if for every compact subset J of Y there exists a compact subset K of X such that
The composition of a bijective proper map followed by a coercive map is coercive.
[edit] Coercive operators and forms
A self-adjoint operator 
where H is a real Hilbert space, is called coercive if there exists a constant c > 0 such that
for all x in H.
A bilinear form 
is called coercive if there exists a constant c > 0 such that
for all x in H.
It follows from the Riesz representation theorem that any symmetric (a(x,y) = a(y,x) for all x,y in H), continuous (
for all x,y in H and some constant K > 0) and coercive bilinear form a has the representation
for some self-adjoint operator which then turns out to be a coercive operator. Also, given a coercive operator self-adjoint operator A, the bilinear form a defined as above is coercive.
One can also show that any self-adjoint operator 
is a coercive operator if and only if it is a coercive function (if one replaces the dot product with the more general inner product in the definition of coercivity of a function). The definitions of coercivity for functions, operators, and bilinear forms are closely related and compatible.
[edit] References
- Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations (Second edition ed.). New York, NY: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0.
- Bashirov, Agamirza E (2003). Partially observable linear systems under dependent noises. Basel; Boston: Birkhäuser Verlag. ISBN 081766999X.
- Gilbarg, David; Trudinger, Neil S. (2001). Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer. ISBN 3540411607.
This article incorporates material from Coercive Function on PlanetMath, which is licensed under the GFDL.
