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 $\varinjlim_i (A_i^*)$