nLab smooth map

Idea

A function which is differentiable function to arbitrary order is called a smooth function.

In the real numbers

Epsilon-delta definition

Let $\mathbb{R}$ be the real numbers. A function $f:\mathbb{R} \to \mathbb{R}$ is smooth if it comes with a sequence of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that

• for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$

• for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,

  $$|(D^{n}f)(x + h) - (D^{n}f)(x) - h (D^{n + 1}f)(x)| \lt \epsilon |h|$$

Unwrapping the recursive definition above, a function $f:\mathbb{R} \to \mathbb{R}$ is smooth if it comes with a sequence of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that

• for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$

• for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,

  $$\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|$$

 Infinitesimal definition

Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A partial function $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$ and nilradical $D$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$

$$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

Equivalently, let $\mathbb{R}[[\epsilon]]$ denote the ring of univariate formal power series on $\mathbb{R}$. $\mathbb{R}[[\epsilon]]$ is a Artinian local $\mathbb{R}$-algebra with homomorphism $h:\mathbb{R} \to \mathbb{R}[[\epsilon]]$. A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all natural numbers $n \in \mathbb{N}$

$$f_A(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

A function $f:I \to \mathbb{R}$ is smooth at an element $a \in I$ if it is smooth at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is smooth if it is smooth at the improper subset of $I$.

Between Cartesian spaces

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).

Between smooth manifolds

A topological manifold whose transition functions are smooth maps is a smooth manifold. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of atlases, hence which is a $k$-fold differentiable function (see there for more details), for all $k$ The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps between them.

Between generalized smooth spaces

There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.

For details see for example at smooth set.

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

• differentiable function

• smooth function with rapidly decreasing derivatives

• smooth homotopy, smooth isotopy

• nonlinear functional, linear functional

• Fabius function

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

References

An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:

• Lev Pontrjagin, Chapter I of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)