On the completed hull of non-degenerate inner product spaces

Abstract

Motivated by applications to general QFT, the completed hulls of non-degenerate inner product spaces $E,(\cdot ,\cdot )$ are considered. In order to decide whether or not $\tilde{E}$ endowed with the continuously extended inner product $(\cdot ,\cdot {)}^{\sim }$ is non-degenerate, the concept of c-linked l.c. topologies $\tau$ is introduced, and it is shown that $\tilde{E},(\cdot ,\cdot {)}^{\sim }$ is non-degenerate if and only if $\tau$ is c-linked to some minimal majorant. Further, the concepts of quasi-decomposable inner product spaces and quasi-decomposition norms are introduced, and it is shown that $\tilde{E},(\cdot ,\cdot {)}^{\sim }$ is a Krein space.