Resolvent Mapping is Analytic/Bounded Linear Operator/Proof 1

Proof
For $a \in \map \rho T$, define:
 * $R_a = \paren {T - a I}^{-1}$

Then we have:

From Resolvent Mapping is Continuous we have:
 * $R_{z + h} \to R_z$ as $h \to 0$

Taking limits of both sides and using Norm is Continuous, we get:


 * $\ds \lim_{h \mathop \to 0} \dfrac {\norm {\map f {z + h} - \map f z - \paren {T - z I}^{-2} h }_*} {\size h} = \norm {R_z^2 - R_z^2}_* = 0$

which is the result.