nLab
smooth map

Definition

A function on (some open subset of) a cartesian space $\mathbb{R}^n$ with values in the real line $\mathbb{R}$ is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if $A \subseteq \mathbb{R}^n$ is any subset, a function $f: A \to \mathbb{R}$ is defined to be smooth if it has a smooth extension to an open subset containing $A$.

By coinduction: A function $f : \mathbb{R} \to \mathbb{R}$ is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

For $A \subseteq \mathbb{R}^n$, a smooth map $\phi: A \to \mathbb{R}^m$ is a function such that $\pi \circ \phi$ is a smooth function for every linear functional $\pi: \mathbb{R}^m \to \mathbb{R}$. (In the case of finite-dimensional codomains as here, it suffices to take the $\pi$ to range over the $m$ coordinate projections.)

The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).

A manifold whose transition functions are smooth maps is a smooth manifold. The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.

Yet more generally, the morphisms between generalised smooth spaces are smooth maps.

For functions between manifolds that fall short of full smoothness, see differentiable map.

Properties

Basic facts about smooth functions are

• the Hadamard lemma

• Borel's theorem

• the Tietze extension theorem

• the Steenrod-Wockel approximation theorem

• embedding of smooth manifolds into formal duals of R-algebras

• derivations of smooth functions are vector fields

Examples

Every analytic functions (for instance a holomorphic function) is also a smooth function.

A crucial property of smooth functions, however, is that they contain also bump functions.

Related concepts

• nonlinear functional, linear functional

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization