On orthogonal properties of immediate extensions of \(c_0\) (Q716492)

Let \(K\) be a complete non-archimedean non-trivially valued field. Recall that a normed space over \(K\) is called \textit{Cartesian} if every finite-dimensional subspace has an orthogonal base; \textit{Hilbertian} if every finite-dimensional (one-dimensional) subspace has an orthogonal complement. It is known that every Hilbertian space is Cartesian and that, if \(K\) is spherically complete, all normed spaces over \(K\) are Hilbertian, hence Cartesian. The question whether every Cartesian space over a non-spherically complete \(K\) is Hilbertian, was an open problem, posed by van Rooij and Schikhof in 1991. In the present paper the author gives a negative answer to this long standing problem. More concretely, he proves the following. Assume \(K\) is not-spherically complete. Let \(\hat{K}\) be the spherical completion of \(K\), let \(\ell^{\infty}_{\hat{K}}\) be the normed space over \(K\) of all bounded maps \(\mathbb{N} \rightarrow \hat{K}\) with the supremum norm, let \(E\) be a maximal immediate extension of \(c_0\) contained in \(\ell^{\infty}\). Then, there exist \(x \in \ell^{\infty}_{\hat{K}}\) such that \(K x + E\) is Cartesian but not Hilbertian.