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 ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are 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.