# Exact sequence of locally convex vector spaces

Let $$K$$ be a non-archimedean field and $$$$A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C\stackrel{h}{\longrightarrow} D \stackrel{i}{\longrightarrow} E$$$$ an exact sequence of $$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)?

• 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. – CJS Nov 12 at 13:12