Hadamard's lemma

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

Let ƒ be a smooth, real-valued function defined on an open, star-convex neighborhood U of a point a in n-dimensional Euclidean space. Then ƒ(x) can be expressed, for all x in U, in the form:

f(x)=f(a)+\sum_{i=1}^n (x_i-a_i) g_i(x),

where each gi is a smooth function on U, a = (a1,...,an), and x = (x1,...,xn).

Proof

Let x be in U. Let h be the map from [0,1] to the real numbers defined by

h(t)=f(a+t(x-a)).\,

Then since

h'(t)=\sum_{i=1}^n \frac{\partial f}{\partial x_i}(a+t(x-a)) (x_i-a_i),

we have

h(1)-h(0)=\int_0^1 h'(t)\,dt =\int_0^1 \sum_{i=1}^n \frac{\partial f}{\partial x_i}(a+t(x-a)) (x_i-a_i)\, dt =\sum_{i=1}^n (x_i-a_i)\int_0^1 \frac{\partial f}{\partial x_i}(a+t(x-a))\, dt.

But additionally, h(1) − h(0) = f(x) − f(a), so if we let

g_i(x)=\int_0^1 \frac{\partial f}{\partial x_i}(a+t(x-a))\, dt,

we have proven the theorem.

References