# 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.

## 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 *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

- Kachurovskii, I. R. (1960). "On monotone operators and convex functionals".
*Uspekhi Mat. Nauk*.**15**(4): 213–215. - Showalter, Ralph E. (1997).
*Monotone operators in Banach space and nonlinear partial differential equations*. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 80. ISBN 0-8218-0500-2. MR1422252 (Proposition 7.4)

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