Definition:Lipschitz Norm
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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 Lipschitz space on $X _\mathbf A$.
The Lipschitz norm on $F_\theta$ is defined as:
- $\forall f \in F_\theta: \norm f_\theta := \norm f_\infty + \size f_\theta$
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.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
Sources
- 1990: William Parry and Mark Pollicott: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics: Chapter $1$: Subshifts of Finite Type and Function Spaces