In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.
Let ƒ be a smooth, real-valued function defined on an open, star-convex neighborhood U of a point a in n-dimensional Euclidean space. Then ƒ(x) can be expressed, for all x in U, in the form:
where each gi is a smooth function on U, a = (a1,...,an), and x = (x1,...,xn).
Let x be in U. Let h be the map from [0,1] to the real numbers defined by
But additionally, h(1) − h(0) = f(x) − f(a), so if we let
we have proven the theorem.