Let $K$ be a non-archimedean field and \begin{equation} A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C\stackrel{h}{\longrightarrow} D \stackrel{i}{\longrightarrow} E \end{equation} an exact sequence of $K$-vector spaces. Furthermore $A$, $D$ are finite dimensional (hence canonically Banach spaces) and $B$, $E$ are Fréchet spaces (resp. Banach spaces). Additionally the topology on $C$ is the locally convex final topology induced by the morphism $B \stackrel{g}{\rightarrow} C$. This implies that the above sequence is topological exact.

Is then $C$ automatically a Fréchet space (resp. Banach space)?

  • $\begingroup$ I found a positive answer, but I have not enough "reputation" to post it. So if someone is interested then one additional upvote is enough. $\endgroup$ – CJS Nov 12 at 13:12

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