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
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.
Consequences and applications
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.
See also
- Bump function – Smooth and compactly supported function
- Continuously differentiable – Mathematical function whose derivative existsPages displaying short descriptions of redirect targets
- Smoothness – Number of derivatives of a function (mathematics)
- Taylor's theorem – Approximation of a function by a truncated power series
Citations
References
- 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.