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[1] — Let be a smooth, real-valued function defined on an open, star-convexneighborhood of a point in -dimensional Euclidean space. Then can be expressed, for all in the form: where each is a smooth function on and
Corollary[1] — If is smooth and then is a smooth function on Explicitly, this conclusion means that the function that sends to is a well-defined smooth function on
Proof
By Hadamard's lemma, there exists some such that so that implies
Corollary[1] — If are distinct points and is a smooth function that satisfies then there exist smooth functions () satisfying for every such that
Proof
By applying an invertibleaffine 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.