In [1-4] Hilbert spaces over valued fields of infinite rank have been studied. In this paper we introduce the wider class of Norm Hilbert spaces (NHS) i.e. Banach spaces of which closed subspaces admit projections of norm [is less than or equal to] 1. We characterize NHS in several ways (Theorem 4.3)., Bounded orthogonal sequences tend to 0 implying that every ball is a compactoid, a property which in rank 1 theory is only shared by finite-dimensional spaces. Finally we describe in Section 5 those NHS for which there exists a Hermitean form (,) satysfying [...] for all x.