nLab
Hadamard lemma

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

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 fC ()f \in C^\infty(\mathbb{R}) on the real line there is a smooth function gg such that

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

This function gg is also called a Hadamard quotient.

Corollary

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

f(x)=f(0)+xf(0)+12x 2f(0)++1n!x nh(x). 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 n\mathbb{R}^n the lemma says that there are for each fC (X)f \in C^\infty(X) nn smooth functions g ig_i such that

f(x)=f(0)+ ix ig 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 ff

g i(0)=fx i(0). g_i(0) = \frac{\partial f}{\partial x_i}(0) \,.
Proof

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

f(x)f(0)= 0 1h(t)dt= 0 1 i=1 nfx i(tx)x idt 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)= 0 1fx i(tx)dt. g_i(x) = \int_{0}^{1} \frac{\partial f}{\partial x_i}(t x) d t \,.

Remarks

References

The Hadamard lemma is due to Jacques Hadamard.

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 April 23, 2014 17:28:29 by Joj? (198.84.250.178)