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 edit
Hadamard's lemma[1] — 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:
Proof edit
Let Define by
Then
But additionally, so by letting
Consequences and applications edit
Corollary[1] — If is smooth and then is a smooth function on Explicitly, this conclusion means that the function that sends to
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
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
See also edit
- Bump function – Smooth and compactly supported function
- Continuously differentiable – Mathematical function whose derivative exists
- Smoothness – Number of derivatives of a function (mathematics)
- Taylor's theorem – Approximation of a function by a truncated power series
Citations edit
- ^ a b c Nestruev 2020, pp. 17–18.
References edit
- Nestruev, Jet (2002). Smooth manifolds and observables. Berlin: Springer. ISBN 0-387-95543-7.
- Nestruev, Jet (10 September 2020). Smooth Manifolds and Observables. Graduate Texts in Mathematics. Vol. 220. Cham, Switzerland: Springer Nature. ISBN 978-3-030-45649-8. OCLC 1195920718.