Let $(E\_n,i\_n)\_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E\_{\infty}$ by their inductive limit. What is the family of semi-norms defining the LCS topology on $E\_{\infty}$?


Similarly, if $(E\_n,\pi\_n)$ is a projective system of Banach spaces in the category of LCS, then what are the semi-norms defining the topology of their projective limit?



Semi-norms on LCS inductive limit of Banach Spaces
There is a simple **abstract** description of the semi-norms of an inductive limit of Banach or locally convex space $E\_n$ with
linking maps $i\_n^m:E\_n\to E\_m$ for $n\le m$ and $i\_n^\infty:E\_n\to E\_\infty$: Just take all semi-norms $p:E\_\infty\to[0,\infty)$ such that $p\circ i\_n^\infty$ is continuous on $E\_n$. In the case of Banach (or normed) spaces $(E\_n,\|\cdot\|\_n)$ this just means $p\circ i\_n^\infty \le c\_n\|\cdot\|\_n$ for some constants $c\_n\ge 0$.


It is quite easy to see that these semi-norms on $E\_\infty$ generate a locally convex topology satisfying the universal property of inductive limits, i.e., a linear map $T:E\_\infty\to X$
into a locally convex space is continuous if and only if all $T\circ i\_n^\infty$ are continuous on $E\_n$.


**However**, If you need to *construct* a semi-norm, e.g., to apply Hahn-Banach, this abstract description is not very helpful. One can try to make the description more concrete, e.g., also the system
of semi-norms
$$
p(x)=\inf\left\{\sum\_{n=1}^N \varepsilon\_n \|x\_n\|\_n: x\_n\in E\_n, x=\sum\_{n=1}^Nx\_n\right\}
$$
with $\varepsilon\_n>0$ generates the inductive limit topology but, apparently, this is not so easy to handle. There are, however, a number of tricks to make constructions of semi-norms feasible. E.g., compactness of the linking maps helps a lot, then you have so-called LS- or DFS-spaces (duals of Fréchet-Schwartz spaces) or *Silva*-spaces. Other tricks (not always stating explicitely the relation to locally convex inductive limits) were developped by Malgrange, Ehrenpreis, and Hörmander in distribution theory.


As you also asked for projective limits: This is much easier. Now you have linear maps $\pi\_\infty^n:F\_\infty\to F\_n$ and the system of semi-norms $p\_n\circ \pi\_\infty^n$ with the given semi-norms on $F\_n$ generates the projective limit topology.


