nLab
Hadamard lemma

This page is about the lemma on Taylor polynomials of smooth functions. For Hadamard's formula in Lie theory see there.

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

The Hadamard lemma says that the Taylor series of a smooth function on the real line around the origin has a remainder at order nn which is the product of x n+1x^{n+1} (for xx the canonical coordinate function) with another smooth function.

Simple as this may sound, it has profound consequences, as it means that smooth functions behave more like polynomials than the classical definition might suggest. For instance the Hadamard lemma directly implies that

  1. derivations of smooth functions are vector fields: For XX a smooth manifold and C (X)C^\infty(X) its \mathbb{R}-algebra of smooth functions (under pointwise multiplication), then there is a natural bijection between the smooth tangent vector fields on XX and the purely algebraic derivations of the algebra C (X)C^\infty(X);

  2. the smooth infinitesimally thickened point 𝔻\mathbb{D} is the same as in algebraic geometry: The quotient of the algebra of smooth functions C ( 1)C^\infty(\mathbb{R}^1) on the real line by the ideal generated by the square x 2x^2 of the canonical coordinate function xx is the ring of dual numbers:

    C ()/(x 2)(ϵ)/(ϵ 2). C^\infty(\mathbb{R})/(x^2) \simeq (\mathbb{R} \oplus \epsilon \mathbb{R})/(\epsilon^2) \,.
  3. together this implies that a tangent vector in a smooth manifold XX is equivalently a morphism of the form

    𝔻X \mathbb{D} \longrightarrow X

    of formal duals of \mathbb{R}-alghebras, from the infinitesimally thickened point 𝔻\mathbb{D}.

This means that differential geometry has more in common with algebraic geometry than is manifest from the traditional definitions. In synthetic differential geometry one makes use of these facts to find a useful unified perspective. For exposition of this point see at geometry of physics -- supergeometry.

More generally one may ask for other types of function algebras which satisfy the conclusion of the Hadamard lemma. These turn out to be the algebras over those algebraic theories which are called Fermat theories. These are hence a crucial ingredient for well-adapted models of synthetic differential geometry.

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 \,.

Related theorems

References

The Hadamard lemma is due to Jacques Hadamard.

Review includes

  • Jet Nestruev, Smooth manifolds and observables, Graduate texts in mathematics 218 (2003)

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 July 6, 2017 06:08:12 by Daniel Luckhardt (134.76.82.216)