Definition:Lipschitz Space

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Let $\struct {X _\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.

Let $\theta \in \openint 0 1$.

The Lipschitz space on $X _\mathbf A$ with respect to the metric $d_\theta$ is defined as:

$\ds \map {F_\theta} {X_\mathbf A} := \set {f \in \map C {X _\mathbf A, \C} : \sup_{n \mathop \in \N} \dfrac {\map {\mathrm {var}_n} f} {\theta^n} < \infty}$


$\map C {X _\mathbf A, \C}$ denotes the continuous mapping space
$\mathrm {var}_n$ denotes the $n$th variation

Also known as

It is also called the space of Lipschitz functions or space of Lipschitz mappings.

In all cases on $\mathsf{Pr} \infty \mathsf{fWiki}$, the term mapping is preferred over function.

If no confusion can arise, the Lipschitz space can also be denoted by $F_\theta$.

It is also written as $\struct {F_\theta, \norm \cdot_\theta}$ together with the Lipschitz norm.

Also see

  • Results about Lipschitz spaces can be found here.

Source of Name

This entry was named for Rudolf Otto Sigismund Lipschitz.