Operators on PIP-Spaces and Indexed PIP-Spaces

Abstract

As already mentioned, the basic idea of pip-spaces is that vectors should not be considered individually, but only in terms of the subspaces V _r (r Є F), the building blocks of the structure. Correspondingly, an operator on a pipspace should be defined in terms of assaying subspaces only, with the proviso that only continuous or bounded operators are allowed. Thus an operator is a coherent collection of continuous operators. We recall that in a nondegenerate pip-space, every assaying subspace V _r carries its Mackey topology $\tau (V_r , V \bar{r})$ and thus its dual is $V \bar{r}$. This applies in particular to $V^{\#}$ and V itself. For simplicity, a continuous linear map between two pip-spaces α : X → Y will always mean a linear map α continuous for the respective Mackey topologies of X and Y. For the sake of generality, it is convenient to define directly operators from one pip-space into another one.