Transfer Operator with respect to One-Sided Shift Space of Finite Type is Linear Bounded Operator

Theorem
Let $\struct {X ^+, \sigma}$ be a one-sided shift of finite type.

Let $\struct {B, \norm \cdot }$ be either:
 * $\struct {\map C {X ^+, \C}, \norm \cdot _\infty}$ the continuous mapping space with the supremum norm
 * or:
 * $\struct {\map {F_\theta^+} {X^+}, \norm \cdot _\theta}$ the space of Lipschitz functions with the Lipschitz norm

Let $f \in B$.

Then the transfer operator $\LL_f : B \to B$ is a bounded linear operator.