Norm Hilbert spaces over Krull valued fields

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 1 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 (Krall) 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 NHS of different nature are presented. Typical results are that NHS are of countable type, that they have orthogonal bases, and that no subspace is linearly homeomorphic to c₀.