The splitting relation for Fréchet spaces over non-archimedean fields

Abstract

Abstract. A complete valued field ( K , | · | ) $(K, |\cdot |)$ is non-archimedean if its valuation satisfies the strong triangle inequality | a + b | ≤ max { | a | , | b | } $|a+b|\le \max \lbrace |a|, |b|\rbrace $ for all a , b ∈ K $a,b \in K$ . 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 ( A s ( a ) , A r ( b ) ) $(A_s(a), A_r(b))$ of power series spaces over 𝕂 is splitting if and only if s = ∞ $s=\infty $ or s = 1 $s=1$ and the set M a , b $M_{a,b}$ of all finite limit points of the double sequence ( a p / b q ) p , q ∈ ℕ $(a_p/b_q)_{p,q\in \mathbb {N}}$ is bounded.