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, ::\big( \mathrm{d} f(x) - \mathrm{d} f(y) \big) (x - y) \geq 0. ==References==