Kachurovskii's theorem

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: f is a convex function; for all x and y in K, {displaystyle mathrm {d} f(x)(y-x)leq f(y)-f(x);} df is an (increasing) monotone operator, i.e., for all x and y in K, {displaystyle {big (}mathrm {d} f(x)-mathrm {d} f(y){big )}(x-y)geq 0.} 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. pp. 80. ISBN 0-8218-0500-2. MR1422252 (Proposition 7.4) hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Convex analysisTheorems in functional analysis