Non-Archimedean Hilbert like spaces

Let K be a non-Archimedean, complete valued field. It is known that the supremum norm ||.||(infinity) on c(0) 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 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.