In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Kachurovskii's theorem is a theorem relating the
convexity of a function on a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
to the
monotonicity
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 orde ...
of its
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued f ...
.
Statement of the theorem
Let ''K'' be a
convex subset
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
of a Banach space ''V'' and let ''f'' : ''K'' → R ∪ be an
extended real-valued function that is Fréchet differentiable with derivative d''f''(''x'') : ''V'' → R at each point ''x'' in ''K''. (In fact, d''f''(''x'') is an element of the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
''V''
∗.) Then the following are equivalent:
* ''f'' is a convex function;
* for all ''x'' and ''y'' in ''K'',
::
* d''f'' is an (increasing) monotone operator, i.e., for all ''x'' and ''y'' in ''K'',
::
References
*
* (Proposition 7.4)
{{Functional analysis
Convex analysis
Theorems in functional analysis