Definition:Lipschitz Norm

From ProofWiki
Jump to navigation Jump to search


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

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

Let $\map {F_\theta} {X _\mathbf A}$ be the Lipschitz space on $X _\mathbf A$.

The Lipschitz norm on $F_\theta$ is defined as:

$\forall f \in F_\theta: \norm f_\theta := \norm f_\infty + \size f_\theta$


$\norm f_\infty$ denotes the supremum norm of $f$
$\size f_\theta$ denotes the Lipschitz seminorm.

Also known as

The Lipschitz norm on $F_\theta$ can also be referred to as the Hölder $C^{\openint 0 1}$ norm, as this is $\norm f_{\map {C^{\openint 0 1} } {F_\theta} }$ on a Hölder space.

Also see

  • Results about the Lipschitz norm can be found here.

Source of Name

This entry was named for Rudolf Otto Sigismund Lipschitz.

Linguistic Note

The term Lipschitz Norm was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.