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 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:

References

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