Definition:Lipschitz Norm

Definition
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 space of Lipschitz mappings.

The Lipschitz norm on $F_\theta$ is defined as:
 * $\forall f \in F_\theta: \norm f_\theta := \norm f_\infty + \size f_\theta$

where:
 * $\norm f_\infty$ 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.