Abstract
Whereas the third paper in this series dealt with the algebraic structure of partial inner product (PIP) spaces, the present one explores systematically their topological properties. A slightly more restricted object is introduced, that we call an indexed PIP-space: it consists of a PIP-space together with a distinguished family of assaying subspaces. The upshot of the analysis is the characterization of two types of indexed PIP-spaces, called type (B) and type (H), respectively, as the most likely candidates for practical applications; they are simply lattices of Banach, resp. Hilbert spaces. Operators on indexed PIP-spaces are discussed and conditions are given that guarantee that the domain of any such operator is a vector subspace. Finally, we examine the question of the existence of a central Hilbert space, in the case of a positive definite partial inner product.
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