Non-Archimedean Hilbert like spaces

Abstract

Let K be a non-Archimedean, complete valued field. It is known that the supremum norm ∥⋅∥∞ on c0 is induced by an inner product if and only if the residual class field of K is formally real. One of the main problems of this inner product is that c0 is not orthomodular, as is any classical Hilbert space. Our goal in this work is to identify those closed subspaces of c0 which have a normal complement. In this study we also involve projections, adjoint and self-adjoint operators.