Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (as TVS) and get: $$ 0\to B\;{\xrightarrow {\ f\ }}\;C\;{\xrightarrow {\ g\ }}\;D\to 0. $$ I know, that direct limit preserves exactness. But I would like to know, new maps will remain continuous?
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$\begingroup$ By definition of an inductive limit, a linear map on the limit is continuous if and only if all compositions with the linking maps from the steps to the limit are continuous. $\endgroup$– Jochen WengenrothCommented Apr 28, 2019 at 13:57
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