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.
Hadamard's lemma — Let be a smooth, real-valued function defined on an open, star-convex neighborhood of a point in -dimensional Euclidean space. Then can be expressed, for all in the form:
where each is a smooth function on and
Proof
Proof
Let Define by
Then
which implies
But additionally, so by letting
the theorem has been proven.
Proof
By applying an invertible affine linear change in coordinates, it may be assumed without loss of generality that and
By Hadamard's lemma, there exist such that
For every let where implies
Then for any
Each of the terms above has the desired properties.