In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Kachurovskii's theorem is a theorem relating the
convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
of a function on a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
to the
monotonicity 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 ...
.
Statement of the theorem
Let ''K'' be a
convex subset 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