Norm Hilbert spaces over Krull valued fields

Abstract

Norm Hilbert spaces (NHS) are defined as Banach spaces over valued fields (see 1.4) for which each closed subspace has a norm-orthogonal complement. For fields with a rank I valuation, these spaces were characterized already in [10, 5.13, 5.16], where it was proved that infinite-dimensional NHS exist only if the valuation of K is discrete. The first discussion of the case of (Krull) valued fields appeared in [1] and [3]. In this paper we continue and expand this work focussing on the most interesting cases, not covered before. If K is not metrizable then each NHS is finite-dimensional (Corollary 3.2.2), but otherwise there do exist infinite-dimensional NHS; they are completely described in 3.2.5. Our main result is Theorem 3.2.1, where various characterizations of NI-IS of different nature are presented. Typical results are that NHS arc of countable type, that they have orthogonal bases, and that no subspace is linearly homeomorphic to co.