functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

, a set-valued mapping

A:X\to 2^X

where ''X'' 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 natu ...

is said to be strongly monotone if :

\exists\,c>0 \mbox \langle u-v , x-y \rangle\geq c \, x-y\, ^2 \quad \forall x,y\in X, u\in Ax, v\in Ay.

This is analogous to the notion of

strictly increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...

for scalar-valued functions of one scalar argument.

References

* Zeidler. ''Applied Functional Analysis'' (AMS 108) p. 173 * Hilbert spaces {{Mathanalysis-stub

See also

References