Hadamard's lemma


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.

Statement

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 can be expressed, for all x in U, in the form:
where each gi is a smooth function on U, a =, and x =.

Proof

Let x be in U. Let h be the map from to the real numbers defined by
Then since
we have
But, additionally, h − h = f − f, so if we let
we have proven the theorem.