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.