Kachurovskii's theorem
Encyclopedia
In mathematics
, Kachurovskii's theorem is a theorem relating the convexity
of a function on a Banach space
to the monotonicity of its Fréchet derivative
.
of a Banach space V and let f : K → R ∪ {+∞} be an extended real-valued function
that is Fréchet differentiable with derivative df(x) : V → R at each point x in K. (In fact, df(x) is an element of the continuous dual space V∗.) Then the following are equivalent:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Kachurovskii's theorem is a theorem relating the convexity
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
of a function on a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
to the monotonicity of its Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
.
Statement of the theorem
Let K be a convex subsetConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
of a Banach space V and let f : K → R ∪ {+∞} be an extended real-valued function
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
that is Fréchet differentiable with derivative df(x) : V → R at each point x in K. (In fact, df(x) is an element of the continuous dual space V∗.) Then the following are equivalent:
- f is a convex function;
- for all x and y in K,
- df is an (increasing) monotone operator, i.e., for all x and y in K,
