A function space integral for a Banach space of functionals on Wiener space

In an earlier paper the authors established the existence of Cameron and Storvick's function space integral ${J_q}(F)$ for a class of finite-dimensional functionals $F$. Here we consider a space $A$ of not necessarily finite-dimensional functionals generated by the earlier functionals. We show that $A$ is a Banach space and recognize $A$ as the direct sum of more familiar Banach spaces. We also show that the function space integral $J_q^{{\text{an}}}(F)$ exists for $F \in A$. In contrast we give an example of an ${F_0} \in A$ such that $J_q^{{\text{seq}}}({F_0})$ does not exist.