On the stability of orthogonal bases in non-archimedean metrizable locally convex spaces (Q5952921)

Let \(E\) be a locally convex Hausdorff space (lcs) over a non-Archimedean non-trivially valued field \(\mathbb{K}\) which is complete under the metric induced by the valuation \(|\cdot|:\mathbb{K}\to [0,\infty)\). An orthogonal basis \((x_n)\) in \(E\) is called stable if there exists a sequence \((u_n)\) of neighbourhoods of the zero in \(E\) such that any sequence \((y_n)\subset E\) with \(y_n\in (x_n+ u_n)\), \(n\in\mathbb{N}\) is an orthogonal basis in \(E\) which is equivalent to \((x_n)\) (sequences \((x_n)\) and \((y_n)\) in \(E\) are equivalent if there exists a linear homeomorphism \(P\) between the linear spans of \((x_n)\) and \((y_n)\) such that \(px_n= y_n\) for all \(n\in\mathbb{N}\)).NEWLINENEWLINENEWLINEIn this article, the problem of the stability of orthogonal bases and basic orthogonal sequences in metrizable locally converse spaces (in particular, Schauder bases and basic sequences in Fréchet spaces) is studied. The main results proved are:NEWLINENEWLINENEWLINE(i) Any orthogonal basis in a metrizable lcs with a continuous norm is stable (Proposition 5(b)); in particular, any Schauder basis in a Fréchet space with a continuous norm is stable (Corollary 6).NEWLINENEWLINENEWLINE(ii) A metrizable lcs without a continuous norm has a stable orthogonal basis (Proposition 5(a)) and if \(E\) is not of finite type, then it contains a dense subspace without an orthogonal basis (Proposition 8(b)). In particular, the Fréchet space \(c_0\times \mathbb{K}^{\mathbb{N}}\) with an orthogonal basis contains a dense subspace without an orthogonal basis (Example a).NEWLINENEWLINENEWLINEIt is still unknown whether any Fréchet space of countable type has an orthogonal basis.