Monotonicity in Functional Analysis
In functional analysis on a topological vector space X, a (possibly non-linear) operator T : X → X∗ is said to be a monotone operator if
Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.
A subset G of X × X∗ is said to be a monotone set if for every pair and in G,
G is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator G(T) is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.
Read more about this topic: Monotonic Function
Famous quotes containing the words functional and/or analysis:
“Indigenous to Minnesota, and almost completely ignored by its people, are the stark, unornamented, functional clusters of concrete—Minnesota’s grain elevators. These may be said to express unconsciously all the principles of modernism, being built for use only, with little regard for the tenets of esthetic design.”
—Federal Writers’ Project Of The Wor, U.S. public relief program (1935-1943)
“Whatever else American thinkers do, they psychologize, often brilliantly. The trouble is that psychology only takes us so far. The new interest in families has its merits, but it will have done us all a disservice if it turns us away from public issues to private matters. A vision of things that has no room for the inner life is bankrupt, but a psychology without social analysis or politics is both powerless and very lonely.”
—Joseph Featherstone (20th century)