Context

Differential geometry

differential geometry

synthetic differential geometry

Contents

Disambiguation

Distinguish from Hadamard's formula in Lie theory, which is also often called Hadamard's lemma.

Statement

Proposition

For every smooth function $f \in C^\infty(\mathbb{R})$ on the real line there is a smooth function $g$ such that

$f(x) = f(0) + x \cdot g(x) \,.$

This function $g$ is also called a Hadamard quotient.

Corollary

It follows that $g(0) = f'(0)$ is the derivative of $f$ at 0. By applying this repeatedly the lemma says that $f$ has a partial Taylor series expansion whose remainder $h$ is a smooth function:

$f(x) = f(0) + x f'(0) + \frac1{2} x^2 f''(0) + \cdots + \frac1{n!} x^n h(x) \,.$

More generally, for smooth functions on any Cartesian space $\mathbb{R}^n$ the lemma says that there are for each $f \in C^\infty(X)$ $n$ smooth functions $g_i$ such that

$f(\vec x) = f(0) + \sum_i x_i g_i(x) \,.$

So at the origin these smooth functions compute the partial derivatives of $f$

$g_i(0) = \frac{\partial f}{\partial x_i}(0) \,.$
Proof

Holding $x$ fixed, put $h(t) = f(t x)$. Then

$f(x) - f(0) = \int_{0}^{1} h'(t) d t = \int_{0}^{1} \sum_{i=1}^n \frac{\partial f}{\partial x_i} (t x) \cdot x_i d t$

where the second equality uses the chain rule. The lemma follows by putting

$g_i(x) = \int_{0}^{1} \frac{\partial f}{\partial x_i}(t x) d t \,.$