nLab
Hadamard lemma

Disambiguation

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

Statement

Proposition

(Hadamard lemma)

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

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

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

Corollary

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

$$f(x)=f(0)+xf\prime (0)+x^{2}f″(0)+\cdots +x^{n}h(x).$$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 $f\in C^{\mathrm{\infty }}(X)$ $n$ smooth functions $g_i$ such that

$$f(\stackrel{\rightharpoonup }{x})=f(0)+\sum _i{x}_i{g}_i(x).$$f(\vec x) = f(0) + \sum_i x_i g_i(x) \,.

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

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

Proof

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

$$f(x)-f(0)={\int }_0^{1}h\prime (t)dt={\int }_0^1\sum _{i=1}^n\frac{\partial f}{\partial x_i}(tx)\cdot x_{i}dt$$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

$$g_i(x)={\int }_0^1\frac{\partial f}{\partial x_i}(tx)dt.$$g_i(x) = \int_{0}^{1} \frac{\partial f}{\partial x_i}(t x) d t \,.

Remarks

• The Hadamard lemma implies that the derivations of the algebra $C^{\mathrm{\infty }}(X)$ of smooth functions on a smooth manifold are in bijection with the vector fields on $X$. See derivation for details.

• A more abstract way to state the Hadamard lemma (and a bit more) is to say that smooth function rings form a Fermat theory. As such the Hadamard lemma is a crucial ingredient for well-adapted models of synthetic differential geometry.

Related theorems

• Borel's theorem

• the Tietze extension theorem

• the Steenrod-Wockel approximation theorem

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

• Ieke Moerdijk, Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis Springer (1991)