nLab Lipschitz map

Idea

Lipschitz maps are functions between metric spaces that lie intermediate between the uniformly continuous maps and the short maps. (That is, every short map is Lipschitz, and every Lipschitz map is uniform.)

One can also consider Lipschitz maps between gauge spaces with appropriate gauges, or between manifolds with appropriate atlases.

Definitions

Let $X$ and $Y$ be metric spaces (with the metric denoted $d$ for both). Let $f\colon X \to Y$ be a function.

Definition

The Lipschitz norm or Lipschitz modulus ${\|f\|_{Lip}}$ of $f\colon X \to Y$ is the supremum of the absolute difference quotients

$$\frac{d(f(a), f(b))} {d(a, b)}$$

for $a \neq b$ in $X$.

In general, this supremum is a lower real number; in particular, it may be infinite. We work in the space $[0, \infty]$ of nonnegative lower reals, so that the Lipschitz norm is $0$ if $X$ is a subsingleton (so that $a \neq b$ never occurs).

Definition

The map $f\colon X \to Y$ is Lipschitz continuous, or simply Lipschitz, if its Lipschitz norm is finite.

We can be more specific:

Definitions

The map $f\colon X \to Y$ is short if its Lipschitz norm is at most $1$. It is a contraction mapping? if its Lipschitz norm is strictly less than $1$.

Properties

In classical mathematics, one calls $K \in [0, \infty]$ a Lipschitz constant of $f\colon X \to Y$ if

$$d(f(a), f(b)) \leq K d(a, b)$$

for all $a, b\colon X$. Then the Lipschitz norm is the minimum of the Lipschitz constants (which is always attained, if there is any finite Lipschitz constant at all). In constructive mathematics, this procedure yields an extended upper real that is (in general) too large. (For example, let $f$ be the sequence of partial sums of a Specker sequence, thought of as a map from $\mathbb{N} \subseteq \mathbb{R}$ to $\mathbb{R}$.)

If you want a definition that is constructively acceptable and yet involves no division, then take the Lipschitz norm of $f$ to be the supremum over those $k$ such that

$$d(f(a), f(b)) \gt k d(a, b)$$

for some $a, b\colon X$. (This is perhaps the best definition, as it will automatically take care of subsingletons, as long as you restrict attention to $[0, \infty]$ a priori, and pseudometric spaces, even constructively.) You can even restrict attention to (say) rational $k$, or to $a, b$ in some dense subspace of $X$ (say if $X$ is separable).

Every Lipschitz map is uniformly continuous. Every Lipschitz map from the real line $\mathbb{R}$ to itself is absolutely continuous.

If a map $f$ between the real line and itself is differentiable, then its Lipschitz norm is the supremum norm ${\|f'\|_\infty}$ of its derivative $f'$. (This result can actually be generalised quite a bit, using Rademacher's theorem? and its generalisations.)

Generalisations

The definitions above generalise immediately to quasimetric spaces; there is no asymmetry between left and right, and the Lipschitz norm of $f\colon X \to Y$ is the same as that of $f^op\colon X^op \to Y^op$ (although the map from $X^op$ to $Y$, equivalently the map from $X$ to $Y^op$, may be quite different).

The definitions also generalise to pseudometric spaces if we take the supremum over those $a \neq b$ such that $d(a, b) \gt 0$. However, we must also require that a Lipschitz map preserve metric equivalence: $d(f(a), f(b)) = 0$ if $d(a, b) = 0$. Then we may pass to the quotient metric spaces and preserve the Lipschitz norm.

Let $X$ be a gauge space, that is a set equipped with a collection of pseudometrics (the gauge of $X$). Then $X$'s gauge is a Lipschitz gauge if the identity function of $X$ is Lipschitz continuous as a map between $X$ equipped with any two of its pseudometrics. Then we may extend $X$'s gauge to a unique maximal Lipschitz gauge and say that $X$ (so equipped) is a Lipschitz gauge space. We may then unambiguously define Lipschitz continuous maps (but not their Lipschitz norms) between any two Lipschitz gauge spaces.

Let $X$ be a topological manifold, and equip $X$ with a compatible atlas whose transition maps are all Lipschitz (a Lipschitz atlas). Then we may extend $X$'s atlas to a unique maximal compatible Lipschitz atlas and say that $X$ (so equipped) is a Lipschitz manifold. We may then unambiguously define Lipschitz continuous maps (but not their Lipschitz norms) between any two Lipschitz manifolds. Assuming that our Lipschitz manifolds are paracompact, then we may think of them as Lipschitz gauge spaces (with gauges consisting of all Lipschitz-continuous pseudometrics), recovering the same notion of Lipschitz map.

A map $f\colon X \to Y$ is Lipschitz of order $\alpha$, or Hölder of order $\alpha$, if the supremum of

$$\frac{d(f(a), f(b))} {d(a, b)^\alpha}$$

is finite. Here, $\alpha$ may be any positive number (but is typically taken to be less than $1$). Everything on this page generalises fairly well to these functions (changing also the definition of derivative to match). Of course, a Lipschitz map of order $\alpha$ from $X$ to $Y$ is simply a Lipschitz map from $X^\alpha$ to $Y$, where $X^\alpha$ is $X$ with all distances raised to the power of $\alpha$.

Lipschitz (and Hölder) maps also have non-uniform versions (fix $a$, but let the norm vary with $b$, which introduces an asymmetry for quasimetric spaces), as well as local versions (every point has a neighbourhood on which $f$ is Lipschitz). I kind of get lost in all of the possibilities.

Related pages

• Lipschitz domain?
• corkscrew domain?
• convex function