Let $E$ be a Banach space, $A_n$ be an increasing sequence of finite dimensional subspaces of $E$, $B_n$ be an increasing sequence of subspaces of $A_n$ and let $C_n = A_n/B_n$. Assume that the dimension of $C_n$ is constant, equal to $d$, and that $A_n \subset B_{n+1} $.
Unless I'm mistaken, in this setting there are natural isomorphisms $j_n : C_n \rightarrow C_{n+1}$ and the associated direct limit is $C=A/B$, where $A= \cup_n A_n$ and $B = \cup_n B_n$, and the dimension of $C$ is $d$. In particular for any vector space topology it is complete.
Question : is there a natural inherited norm in this setting on $C$ ? I would think there is such a norm on $C' = \overline{A}/\overline{B}$, but it doesn't seem true that $C'$ is isomorphic to $C$ in general (or is it ?)
Maybe there is a condition on $\|j_n\|$ for such a norm to exist ?