General Theory: Algebraic Point of View

Abstract

It is convenient to divide our study of pip-spaces into two stages. In the first one, we consider only the algebraic aspects. That is, we explore the structure generated by a linear compatibility relation on a vector space V , as introduced in Section I.2, without any other ingredient. This will lead us to another equivalent formulation, in terms of particular coverings of V by families of subspaces. This first approach, purely algebraic, is the subject matter of the present chapter. Then, in a second stage, we introduce topologies on the so-called assaying subspaces $\{V_r \}$. Indeed, as already mentioned in Section I.2, assuming the partial inner product to be nondegenerate implies that every matching pair $(V_r , V \bar{r})$ of assaying subspaces is a dual pair in the sense of topological vector spaces. This in turn allows one to consider various canonical topologies on these subspaces and explore the consequences of their choice. These considerations will be developed at length in Chapter 2.