Non-Archimedean Hilbert like spaces

Abstract

Let $\mathbb{K}$ be a non-Archimedean, complete valued field. It is known that the supremum norm $\left\Vert \cdot\right\Vert _{\infty}$ on $c_{0}$ is induced by an inner product if and only if the residual class field of $\mathbb{K}$ is formally real. One of the main problems of this inner product is that $c_{0}$ is not orthomodular, as is any classical Hilbert space. Our goal in this work is to identify those closed subspaces of $c_{0}$ which have a normal complement. In this study we also involve projections, adjoint and self-adjoint operators.