Kachurovskii's theorem

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.

tatement 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 "V"^∗.) Then the following are equivalent:

* "f" is a convex function;

* for all "x" and "y" in "K",

::$mathrm\{d\}\; f(x)\; (y\; -\; x)\; leq\; f(y)\; -\; f(x);$

* d"f" is an (increasing) monotone operator, i.e., for all "x" and "y" in "K",

::

References

* cite book
last = Showalter
first = Ralph E.
title = Monotone operators in Banach space and nonlinear partial differential equations
series = Mathematical Surveys and Monographs 49
publisher = American Mathematical Society
location = Providence, RI
year = 1997
pages = 80
isbn = 0-8218-0500-2 MathSciNet|id=1422252 (Proposition 7.4)