Coercive function

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.

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. \frac{f(x) \cdot x}{\| x \|} \to + \infty \text{ as } \| x \| \to + \infty, where " \cdot " denotes the usual dot product and \|x\| denotes the usual Euclidean norm of the vector x. A coercive vector field is in particular norm-coercive since \|f(x)\| \geq (f(x) \cdot x) / \| x \| for x \in \mathbb{R}^n \setminus \{0\} , by Cauchy–Schwarz inequality. 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 f(x) \cdot x = 0 for every x \in \mathbb{R}^2 . ==Coercive operators and forms== A self-adjoint operator A:H\to H, where H is a real Hilbert space, is called coercive if there exists a constant c>0 such that \langle Ax, x\rangle \ge c\|x\|^2 for all x in H. A bilinear form a:H\times H\to \mathbb R is called coercive if there exists a constant c>0 such that a(x, x)\ge c\|x\|^2 for all x in H. It follows from the Riesz representation theorem that any symmetric (defined as a(x, y)=a(y, x) for all x, y in H ), continuous ( |a(x, y)|\le k\|x\|\,\|y\| for all x, y in H and some constant k>0 ) and coercive bilinear form a has the representation a(x, y)=\langle Ax, y\rangle for some self-adjoint operator A:H\to H, which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator A, the bilinear form a defined as above is coercive. If 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, \langle Ax, x\rangle \ge C\|x\| for big \|x\| (if \|x\| is bounded, then it readily follows); then replacing x by x\|x\|^{-2} we get that A is a coercive operator. One can also show that the converse holds true if 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 \to X' between two normed vector spaces (X, \| \cdot \|) and (X', \| \cdot \|') is called norm-coercive if and only if \|f(x)\|' \to + \infty \mbox{ as } \|x\| \to +\infty . More generally, a function f : X \to X' between two topological spaces X and X' is called coercive if for every compact subset K' of X' there exists a compact subset K of X such that f (X \setminus K) \subseteq X' \setminus K'. The composition of a bijective proper map followed by a coercive map is coercive. ==(Extended valued) coercive functions== An (extended valued) function f: \mathbb{R}^n \to \mathbb{R} \cup \{- \infty, + \infty\} is called coercive if f(x) \to + \infty \mbox{ as } \| x \| \to + \infty. A real valued coercive function f:\mathbb{R}^n \to \mathbb{R} is, in particular, norm-coercive. However, a norm-coercive function f:\mathbb{R}^n \to \mathbb{R} is not necessarily coercive. For instance, the identity function on \mathbb{R} is norm-coercive but not coercive. ==References==

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