Assume $(A\_i)\_{i \in I}$ is a family of locally convex topological
vector spaces which are all moreover assumed to be Banach spaces.


We assume moreover that $(A\_i)\_{i \in I}$ has additional
structure of an inductive system, that is $I$ is a directed pre-ordered set and
for all $i < j$ there are compatible continuous linear maps
$m\_{ij}: A\_i \to A\_j$ and the indictive limit endowed with
induced topology is defined by $\varinjlim\_i A\_i$.


Let now build the dual spaces $A^\*\_i$ of $A\_i$ with inherited
topology. If the family
$(A\_i)\_{i \in I}$ is an inductive system, then $(A\_i^\*)\_{i \in I}$
becomes naturally a projective system with induced maps
$m\_{ij}^\*: A\_j^\* \to A\_i^\*$.


Now the question is how close are the speces $(\varinjlim\_i A\_i)^\*$
and $\varprojlim\_i (A\_i^\*)$ related to each other?


Are they isomorphic and in which category? Clearly, the algebraic aspects
may behave well, that is it becomes surely an isomorphism of vector
spaces. What can we say about
the topological aspects? Is it a homeomorphism? I haven't an conterexample
but I guess that under general assumptions as above not.


If we primary interested in study spaces of bounded operators $A\_i$ with strong operator topology, are there any criteria when $(\varinjlim\_i A\_i)^\*$
and $\varprojlim\_i (A\_i^\*)$ are homeomorphic?


What about if we "reverse the arrows" everywhere in our question,
that is if we start with a *projective system* $(A\_i)\_{i \in I}$ and
want to know
how are $(\varprojlim\_i A\_i)^\*$
and $\varinjlim\_i (A\_i^\*)$



Compatibility of inductive and projective limits with dualization in functional analysis
Here are some remarks on the inductive limit case:


1. The dual of the inductive limit is ALWAYS identifiable (as a vector space) with the projective limit of the duals. This is just the universal property of the inductive limit;
2. The question of whether they are isomorphic as locally convex spaces is a much more delicate question which was investigated in considerable detail in the heyday of lcs theory;
3. One reason for the difficulty is that thereare various candidates for the locally convex topology on the dual of a dual;
4. The result is false in full generality but there are a number of special situations where it does hold for the so-called strong topology (Silva spaces, inductive limits with partitions of unity,...);
5. If one moves away from regarding the dual of a lcs as again a lcs one obtains more pleasing results. Thus if one takes the dual of a lcs to be a convex bornological space (Buchwalter, Hogbe-Nlend) using the equicontinuous sets as the appropriate objects, then a suitable result holds. It is, of course, a matter of taste whether one regards this as a valid approach or as a cop-out;
6. As stated above, there is a copious literature on this topic, both primary and secondary. Unfortunately, a covid lockdown is keeping me away from my institute library so that I am not really in a position to provide more details;
7. The situation with projective limits is analogous. Under suitable conditions (one needs your assumption that the component spaces be Banach spaces plus density conditions), the result is true in the category of vector spaces. If one wants to regard the case of isomorphism in the sense of lcs’s there are special situations where it is true and these have been studied in some detail.


