Exact subcategories of the category of locally convex spaces

In this paper we present a characterization whether the restriction \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {E}^{\prime }:=\lbrace (f,g)\in \mathcal {E}\,\,|\,\,f,g\in \mbox{Mor}({\mathcal {C}}^{\prime })\rbrace$\end{document} of the exact structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {E}$\end{document} of an exact category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$({\mathcal {C}},\mathcal {E})$\end{document} in the sense of Quillen on a full additive subcategory \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {C}}^{\prime }$\end{document} of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {C}}$\end{document} is again an exact structure. We apply our characterization to the exact structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {E}^{top}_{\mbox{LCS}}$\end{document} of short topologically exact sequences in the quasi‐abelian category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mbox{LCS}$\end{document} of locally convex spaces and subcategories thereof.