Linear homeomorphisms of non-classical Hilbert spaces

Abstract

Let K be a complete infinite rank valued field, In [4] we studied Norm Hilbert Spaces (NHS) over K i.e, K-Banach spaces for which closed subspaces admit projections of norm less than or equal to 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms ale bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together - in real or complex theory shared only by finite-dimensional spaces - show that NHS are more 'rigid' than classical Hilbert spaces.