General Theory: Topological Aspects

Abstract

In Chapter 1, we have analyzed the structure of pip-spaces from the algebraic point of view only, (i.e., the compatibility relation). Here we will discuss primarily the topological structure given by the partial inner product itself. The aim is to tighten the definitions so as to eliminate as many pathologies as possible. The picture that emerges is reassuringly simple: Only two types of pip-spaces seem sufficiently regular to have any practical use, namely lattices of Hilbert spaces (LHS) or Banach spaces (LBS), that we have introduced briefly in the Introduction. Our standard reference on locally convex topological vector spaces (LCS) will be the textbook of Köthe [Köt69]. In addition, for the convenience of the reader, we have collected in Appendix B most of the necessary, but not so familiar, notions needed in the text. Notice that we diverge from [Köt69] for the notation. Namely, for a given dual pair $<E, F>$, we denote the weak topology on E by σ(E,F), its Mackey topology by t(E,F), and its strong topology by β(E,F) (see Appendix B).