Equivalent Norms on Lipschitz Space/Shift of Finite Type

Theorem
Let $\struct {X, \sigma}$ be a shift of finite type.

Let $F_\theta$ be the space of Lipschitz mappings on $X$.

Let $x_0 \in F_\theta$.

Then the following norms on $F_\theta$ are equivalent:


 * $(1): \quad$ Lipschitz norm $\norm\cdot_\theta$
 * $(2): \quad \forall f \in F_\theta : \norm f '_\theta := \cmod {\map f {x_0} } + \size f_\theta$