The splitting relation for fréchet spaces over non-archimedean fields

Abstract

A complete valued field (K,|.|) is non-archimedean if its valuation satisfies the strong triangle inequality |a + b| ≤ max{|a|,|b|} for all a; b ε K. We say that a pair .E; F / of Fréchet spaces over a non-archimedean field K is splitting if for every Fréchet space G over K and for every closed subspace D of G such that D is isomorphic to F and G/D is isomorphic to E, we can infer that the subspace D is complemented in G. In this paper we study when a pair .E; F / of Fréchet spaces of countable type over K is splitting. In particular, we show that a pair (As(a),A r (b)) of power series spaces over K is splitting if and only if s = ∞ or s = 1 and the set Ma,b of all finite limit points of the double sequence (ap=bq)p,qεN is bounded. © de Gruyter 2014.