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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 $\varinjlim_i (A_i^*)$

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    $\begingroup$ Perhaps, you heard about stereotype spaces ncatlab.org/nlab/show/stereotype+space In this category the duals to injective limits are projective limits and vice versa. But to have this duality one needs to correct the definition of injective and projective limits in such a way that they become objects in this category. This is however not a restrictive correction. $\endgroup$ – Sergei Akbarov Nov 11 '20 at 16:57
  • $\begingroup$ I am not convinced that the underlying vector spaces of dual-of-inductive limit and projective-limit-of-dual will always coincide. The example I have in mind is to take $A_n = \ell_1^n$ and take $A_n \to A_{n+1}$ to be the embedding $(x_1,\dots,x_n) \mapsto (x_1,\dots, x_n,0)$ $\endgroup$ – Yemon Choi Nov 11 '20 at 18:37
  • $\begingroup$ @YemonChoi: It is well known that $(l_1^k)^*= l_{\infty}^k$ and your claim is that $(\varinjlim_k l_1^k)^* \not \cong \varprojlim_k l_{\infty}^k$ already as abstract vector spaces, right? why, could you sketch an argument? $\endgroup$ – Isak the XI Nov 12 '20 at 21:52
  • $\begingroup$ [deleted some erroneous comments] Sorry, my previous comment is not correct, or at least is not a counter-example. I got mixed up with the "reversed question" of whether the canonical map from the inductive limit of the duals to the dual of the projective limit is an isomorphism, and my claim is that in general this canonical map need not be surjective as a linear map between vector spaces. $\endgroup$ – Yemon Choi Nov 13 '20 at 14:45
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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.

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