# nLab Hadamard lemma

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

$f\left(x\right)=f\left(0\right)+x\cdot g\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$f(x) = f(0) + x \cdot g(x) \,.

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

###### Corollary

It follows that $g\left(0\right)=f\prime \left(0\right)$ 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:

$f\left(x\right)=f\left(0\right)+xf\prime \left(0\right)+{x}^{2}f″\left(0\right)+\cdots +{x}^{n}h\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$f(x) = f(0) + x f'(0) + x^2 f''(0) + \cdots + x^n h(x) \,.

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

$f\left(\stackrel{⇀}{x}\right)=f\left(0\right)+\sum _{i}{x}_{i}{g}_{i}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$f(\vec x) = f(0) + \sum_i x_i g_i(x) \,.

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

${g}_{i}\left(0\right)=\frac{\partial f}{\partial {x}_{i}}\left(0\right)\phantom{\rule{thinmathspace}{0ex}}.$g_i(0) = \frac{\partial f}{\partial x_i}(0) \,.
###### Proof

Holding $x$ fixed, put $h\left(t\right)=f\left(tx\right)$. Then

$f\left(x\right)-f\left(0\right)={\int }_{0}^{1}h\prime \left(t\right)dt={\int }_{0}^{1}\sum _{i=1}^{n}\frac{\partial f}{\partial {x}_{i}}\left(tx\right)\cdot {x}_{i}dt$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

${g}_{i}\left(x\right)={\int }_{0}^{1}\frac{\partial f}{\partial {x}_{i}}\left(tx\right)dt\phantom{\rule{thinmathspace}{0ex}}.$g_i(x) = \int_{0}^{1} \frac{\partial f}{\partial x_i}(t x) d t \,.

## References

The Hadamard lemma is what makes the standard convenient models for synthetic differential geometry tick. Its role in this respect can be seen from proposition 1.2 on in

Revised on December 4, 2012 12:14:04 by Toby Bartels (64.134.165.128)