# Coercive function

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Mathematical function

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In [mathematics](/source/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 f ( x ) ⋅ x ‖ x ‖ → + ∞ as ‖ x ‖ → + ∞ , {\displaystyle {\frac {f(x)\cdot x}{\|x\|}}\to +\infty {\text{ as }}\|x\|\to +\infty ,} where " ⋅ {\displaystyle \cdot } " denotes the usual [dot product](/source/Dot_product) and ‖ x ‖ {\displaystyle \|x\|} denotes the usual Euclidean [norm](/source/Norm_(mathematics)) of the vector *x*.

A coercive [vector field](/source/Vector_field) is in particular norm-coercive since ‖ f ( x ) ‖ ≥ ( f ( x ) ⋅ x ) / ‖ x ‖ {\displaystyle \|f(x)\|\geq (f(x)\cdot x)/\|x\|} for x ∈ R n ∖ { 0 } {\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}} , by [Cauchy–Schwarz inequality](/source/Cauchy%E2%80%93Schwarz_inequality). 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 f ( x ) ⋅ x = 0 {\displaystyle f(x)\cdot x=0} for every x ∈ R 2 {\displaystyle x\in \mathbb {R} ^{2}} .

## Coercive operators and forms

A [self-adjoint operator](/source/Self-adjoint_operator) A : H → H , {\displaystyle A:H\to H,} where H {\displaystyle H} is a real [Hilbert space](/source/Hilbert_space), is called **coercive** if there exists a constant c > 0 {\displaystyle c>0} such that ⟨ A x , x ⟩ ≥ c ‖ x ‖ 2 {\displaystyle \langle Ax,x\rangle \geq c\|x\|^{2}} for all x {\displaystyle x} in H . {\displaystyle H.}

A [bilinear form](/source/Bilinear_form) a : H × H → R {\displaystyle a:H\times H\to \mathbb {R} } is called **coercive** if there exists a constant c > 0 {\displaystyle c>0} such that a ( x , x ) ≥ c ‖ x ‖ 2 {\displaystyle a(x,x)\geq c\|x\|^{2}} for all x {\displaystyle x} in H . {\displaystyle H.}

It follows from the [Riesz representation theorem](/source/Riesz_representation_theorem) that any symmetric (defined as a ( x , y ) = a ( y , x ) {\displaystyle a(x,y)=a(y,x)} for all x , y {\displaystyle x,y} in H {\displaystyle H} ), continuous ( | a ( x , y ) | ≤ k ‖ x ‖ ‖ y ‖ {\displaystyle |a(x,y)|\leq k\|x\|\,\|y\|} for all x , y {\displaystyle x,y} in H {\displaystyle H} and some constant k > 0 {\displaystyle k>0} ) and coercive bilinear form a {\displaystyle a} has the representation a ( x , y ) = ⟨ A x , y ⟩ {\displaystyle a(x,y)=\langle Ax,y\rangle }

for some self-adjoint operator A : H → H , {\displaystyle A:H\to H,} which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator A , {\displaystyle A,} the bilinear form a {\displaystyle a} defined as above is coercive.

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

## Norm-coercive mappings

A mapping f : X → X ′ {\displaystyle f:X\to X'} between two normed vector spaces ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} and ( X ′ , ‖ ⋅ ‖ ′ ) {\displaystyle (X',\|\cdot \|')} is called **norm-coercive** if and only if ‖ f ( x ) ‖ ′ → + ∞ as ‖ x ‖ → + ∞ . {\displaystyle \|f(x)\|'\to +\infty {\mbox{ as }}\|x\|\to +\infty .}

More generally, a function f : X → X ′ {\displaystyle f:X\to X'} between two [topological spaces](/source/Topological_space) X {\displaystyle X} and X ′ {\displaystyle X'} is called **coercive** if for every [compact subset](/source/Compact_space) K ′ {\displaystyle K'} of X ′ {\displaystyle X'} there exists a compact subset K {\displaystyle K} of X {\displaystyle X} such that f ( X ∖ K ) ⊆ X ′ ∖ K ′ . {\displaystyle f(X\setminus K)\subseteq X'\setminus K'.}

The [composition](/source/Function_composition) of a [bijective](/source/Bijection) [proper map](/source/Proper_map) followed by a coercive map is coercive.

## (Extended valued) coercive functions

An (extended valued) function f : R n → R ∪ { − ∞ , + ∞ } {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} \cup \{-\infty ,+\infty \}} is called **coercive** if f ( x ) → + ∞ as ‖ x ‖ → + ∞ . {\displaystyle f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .} A real valued coercive function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is, in particular, norm-coercive. However, a norm-coercive function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is not necessarily coercive. For instance, the [identity function](/source/Identity_function) on R {\displaystyle \mathbb {R} } is norm-coercive but not coercive.

## See also

- [Radially unbounded functions](/source/Radially_unbounded_function)

- [Lax-Milgram lemma](/source/Lax-Milgram_lemma)

## References

- Renardy, Michael; Rogers, Robert C. (2004). *An introduction to partial differential equations* (Second ed.). New York, NY: Springer-Verlag. pp. xiv+434. [ISBN](/source/ISBN_(identifier)) [0-387-00444-0](https://en.wikipedia.org/wiki/Special:BookSources/0-387-00444-0).

- Bashirov, Agamirza E (2003). *Partially observable linear systems under dependent noises*. Basel; Boston: Birkhäuser Verlag. [ISBN](/source/ISBN_(identifier)) [0-8176-6999-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-6999-X).

- Gilbarg, D.; [Trudinger, N.](/source/Neil_Trudinger) (2001). *Elliptic partial differential equations of second order, 2nd ed*. Berlin; New York: Springer. [ISBN](/source/ISBN_(identifier)) [3-540-41160-7](https://en.wikipedia.org/wiki/Special:BookSources/3-540-41160-7).

*This article incorporates material from Coercive Function on [PlanetMath](/source/PlanetMath), which is licensed under the [Creative Commons Attribution/Share-Alike License](https://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA).*

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