Contour integral commutes with bounded linear maps

Given: A unital complex Banach algebra $A$, a complex Banach space $E$, a bounded linear $B:A\to E$, a piecewise $C^1$ contour $\gamma :[a,b]\to \mathbb{C}$ with trace ${\gamma }^{\ast }$ and length $L(\gamma )$, a $C^1$ subdivision $a=t_0<\cdots <t_m=b$ with derivative extensions $v_k$ on the closed pieces, and a continuous $f:{\gamma }^{\ast }\to A$.

${\int }_{\gamma }fdz$ is the limit, over tagged partitions refining the subdivision, of ${\sum }_{j}f(\gamma ({\xi }_j))v_{k(j)}({\xi }_j){\mathrm{\Delta }}_j$, where the derivative extension belonging to the subinterval is used even when a tag is a corner; the chain version is the corresponding finite sum (Banach algebra valued contour integral).

$B$ is complex-linear and bounded, and its operator norm satisfies $B(\lambda u+\mu v)=\lambda B(u)+\mu B(v)$ and $\mathrm{\parallel }B(u)\mathrm{\parallel }\le \mathrm{\parallel }B\mathrm{\parallel }\mathrm{\parallel }u\mathrm{\parallel }$ for all $u,v\in A$ and scalars $\lambda ,\mu$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

For a piecewise $C^1$ path $\gamma$ the length is the sum of the speed integrals over a $C^1$ subdivision: $L(\gamma )={\sum }_k{\int }_{t_{k-1}}^{t_k}\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }dt$; on each such interval the speed is continuous, and the corresponding refined Riemann sums converge to this sum (A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces).